2838:
31:
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2428:
12219:
1217:. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or
2971:
2833:{\displaystyle {\begin{pmatrix}y_{1}^{(n)}\\y_{2}^{(n)}\\\vdots \\y_{m}^{(n)}\end{pmatrix}}={\begin{pmatrix}f_{1}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\\f_{2}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\\\vdots \\f_{m}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\end{pmatrix}}}
9422:
6206:
4297:, who studied them in the mid-1800s. SLPs have an infinite number of eigenvalues, and the corresponding eigenfunctions form a complete, orthogonal set, which makes orthogonal expansions possible. This is a key idea in applied mathematics, physics, and engineering. SLPs are also useful in the analysis of certain partial differential equations.
8628:
3292:{\displaystyle {\begin{pmatrix}f_{1}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\\f_{2}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\\\vdots \\f_{m}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots \\0\end{pmatrix}}}
5598:
1178:). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example
12280:
differential equation, and then plug the solution into the differential equation to validate if it satisfies the equation. If it does then we have a particular solution to the DE, otherwise we start over again and try another guess. For instance we could guess that the solution to a DE has the form:
4228:
Lie's group theory of differential equations has been certified, namely: (1) that it unifies the many ad hoc methods known for solving differential equations, and (2) that it provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations.
12322:
In the case of a first order ODE that is non-homogeneous we need to first find a solution to the homogeneous portion of the DE, otherwise known as the associated homogeneous equation, and then find a solution to the entire non-homogeneous equation by guessing. Finally, we add both of these solutions
4197:. As the latter can be classified according to the properties of the fundamental curve that remains unchanged under a rational transformation, Clebsch proposed to classify the transcendent functions defined by differential equations according to the invariant properties of the corresponding surfaces
9190:
12279:
When all other methods for solving an ODE fail, or in the cases where we have some intuition about what the solution to a DE might look like, it is sometimes possible to solve a DE simply by guessing the solution and validating it is correct. To use this method, we simply guess a solution to the
6926:
4166:
was the first to appreciate the importance of this view. Thereafter, the real question was no longer whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or
3902:
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions can't be analytical
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6002:
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9900:
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4912:
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in order to call this an implicit ODE ; an implicit ODE system satisfying this
Jacobian non-singularity condition can be transformed into an explicit ODE system. In the same sources, implicit ODE systems with a singular Jacobian are termed
971:
9166:
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8732:
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7375:
7204:
12184:
9417:{\displaystyle {\begin{aligned}F(x,y)=&\int ^{x}\mu (\lambda ,y)M(\lambda ,y)\,d\lambda +\int ^{y}Y(\lambda )\,d\lambda \\=&\int ^{y}\mu (x,\lambda )N(x,\lambda )\,d\lambda +\int ^{x}X(\lambda )\,d\lambda =C\end{aligned}}}
9067:
5913:
3444:(DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems. Presumably for additional derivatives, the
8423:
7690:
6689:
1925:
8172:
7881:
1407:
5128:
4125:
was a subject of research from the time of
Leibniz, but only since the middle of the nineteenth century has it received special attention. A valuable but little-known work on the subject is that of Houtain (1854).
6949:
10989:
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1803:
10836:
11562:
3697:
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12534:
6264:
12328:
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5672:
but this would lead to a domain that is not an interval, so that the side opposite to the initial condition would be disconnected from the initial condition, and therefore not uniquely determined by it.
3903:
functions on the whole real line, and because they will be non-Lipschitz functions at their ending time, they are not included in the uniqueness theorem of solutions of
Lipschitz differential equations.
1670:
10730:
10316:
5053:
3549:
2277:
11874:
5766:
11258:
2853:
8623:{\displaystyle {\begin{aligned}F(x,y)&=\int ^{x}M(\lambda ,y)\,d\lambda +\int ^{y}Y(\lambda )\,d\lambda \\&=\int ^{y}N(x,\lambda )\,d\lambda +\int ^{x}X(\lambda )\,d\lambda =C\end{aligned}}}
11895:
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4463:
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9429:
9195:
8854:
8438:
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7896:
7474:
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7114:
6694:
5593:{\displaystyle {\begin{cases}\mathbb {R} &y_{0}=0\\\left(-\infty ,x_{0}+{\frac {1}{y_{0}}}\right)&y_{0}>0\\\left(x_{0}+{\frac {1}{y_{0}}},+\infty \right)&y_{0}<0\end{cases}}}
11456:
9909:
9758:
10111:
7891:
5167:
4985:
3819:
8849:
8205:
6201:{\displaystyle {\begin{array}{rcl}y_{1}'&=&y_{2}\\y_{2}'&=&y_{3}\\&\vdots &\\y_{n-1}'&=&y_{n}\\y_{n}'&=&F(x,y_{1},\ldots ,y_{n}).\end{array}}}
1143:
8737:
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4003:
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In their basic form both of these theorems only guarantee local results, though the latter can be extended to give a global result, for example, if the conditions of
9093:
1085:
13628:
7280:
7109:
1983:
2200:
3305:
1194:
17:
9012:
6921:{\displaystyle {\begin{aligned}P_{1}(x)Q_{1}(y)+P_{2}(x)Q_{2}(y)\,{\frac {dy}{dx}}&=0\\P_{1}(x)Q_{1}(y)\,dx+P_{2}(x)Q_{2}(y)\,dy&=0\end{aligned}}}
4773:
When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely:
13050:
Elementary
Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986,
12478:
13078:
Mathematical
Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M. R. Spiegel, J. Liu, Schaum's Outline Series, 2009, ISC_2N 978-0-07-154855-7
4213:'s work put the theory of differential equations on a better foundation. He showed that the integration theories of the older mathematicians can, using
8368:
7611:
4749:
solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on
809:
14023:
4130:(from 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field worked by various writers, notably
14028:
3882:
solution (a general solution of the homogeneous ODE), which then forms a general solution of the original ODE. This is the terminology used in the
728:
8063:
7798:
1320:
13891:
13106:
Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge
University Press, 2010, ISC_2N 978-0-521-86153-3
12240:
469:
11798:
4252:
are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its
14018:
13351:
7091:{\displaystyle \int ^{x}{\frac {P_{1}(\lambda )}{P_{2}(\lambda )}}\,d\lambda +\int ^{y}{\frac {Q_{2}(\lambda )}{Q_{1}(\lambda )}}\,d\lambda =C}
5060:
11152:{\displaystyle y_{c}=C_{1}e^{-{\frac {x}{2}}\,\left(b+{\sqrt {b^{2}-4c}}\right)}+C_{2}e^{-{\frac {x}{2}}\,\left(b-{\sqrt {b^{2}-4c}}\right)}}
4189:. Collet was a prominent contributor beginning in 1869. His method for integrating a non-linear system was communicated to Bertrand in 1868.
6463:{\displaystyle \mathbf {y} =(y_{1},\ldots ,y_{n}),\quad \mathbf {F} (x,y_{1},\ldots ,y_{n})=(y_{2},\ldots ,y_{n},F(x,y_{1},\ldots ,y_{n})).}
5801:
13621:
4277:
Sturm–Liouville theory is a theory of a special type of second-order linear ordinary differential equation. Their solutions are based on
10740:
13678:
12436:, an open-source application that uses a Python-like syntax with a wide range of capabilities spanning several branches of mathematics.
12356:{\displaystyle {\text{general solution}}={\text{general solution of the associated homogeneous equation}}+{\text{particular solution}}}
11470:
212:
6478:
Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here.
4385:
14147:
12584:
Karras, Tero; Aittala, Miika; Aila, Timo; Laine, Samuli (2022). "Elucidating the Design Space of
Diffusion-Based Generative Models".
6604:
in general). The differential equations are in their equivalent and alternative forms that lead to the solution through integration.
4263:
Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines.
4138:. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.
2390:{\displaystyle \mathbf {y} ^{(n)}=\mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)}
1832:
1708:
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2000:
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343:
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10178:
4994:
3912:
2947:{\displaystyle \mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)}\right)={\boldsymbol {0}}}
1599:
6217:
13614:
12679:
Denis, Byakatonda (2020-12-10). "An
Overview of Numerical and Analytical Methods for solving Ordinary Differential Equations".
5704:
13835:
11176:
384:
13729:
13489:
13410:
13367:
13304:
13018:
12990:
12900:
12879:
12856:
12663:
12528:
274:
13598:
A tutorial on how to model a physical system described by ODE using Scilab standard programming language by
Openeering team.
5610:
5314:
3473:
14013:
721:
176:
13923:
10119:
3878:
can also refer to any solution of the ODE (not necessarily satisfying the initial conditions), which is then added to the
13938:
13719:
12493:
12208:
11723:
10489:
7214:
3887:
595:
249:
13286:
12473:
10007:{\displaystyle x=\pm \int ^{y}{\frac {d\lambda }{\sqrt {2\int ^{\lambda }F(\varepsilon )\,d\varepsilon +C_{1}}}}+C_{2}}
3871:
is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.
12458:, an open source computational environment primarily intended for statistics, which includes packages for ODE solving.
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13688:
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13510:
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13250:
13208:
13190:
13161:
13139:
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13055:
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10913:
4150:. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the
2164:
1158:
Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most
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varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the
338:
257:
232:
8043:{\displaystyle {\begin{aligned}yM(xy)+xN(xy)\,{\frac {dy}{dx}}&=0\\yM(xy)\,dx+xN(xy)\,dy&=0\end{aligned}}}
14279:
14206:
13754:
13278:
10046:
5415:
4371:
4131:
3859:
is derived from the general solution by setting the constants to particular values, often chosen to fulfill set '
3452:, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.
3441:
1935:
1810:
714:
9000:{\displaystyle {\begin{aligned}M(x,y){\frac {dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}}
8356:{\displaystyle {\begin{aligned}M(x,y){\frac {dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}}
4630:
14368:
14069:
13569:
12244:
12026:{\displaystyle y_{c}=\sum _{j=1}^{n}\left(\sum _{\ell =1}^{k_{j}}C_{j,\ell }x^{\ell -1}\right)e^{\alpha _{j}x}}
5140:
4480:
1681:
649:
13825:
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4154:
th degree, so it was the hope of analysts to find a general method for integrating any differential equation.
3766:
374:
14194:
14140:
13840:
13683:
13673:
13540:
13481:
8842:
6607:
In the integral solutions, λ and ε are dummy variables of integration (the continuum analogues of indices in
4122:
766:
389:
217:
207:
9661:{\displaystyle X(x)=M(x,y)-{\frac {\partial }{\partial x}}\int ^{y}\mu (x,\lambda )N(x,\lambda )\,d\lambda }
9540:{\displaystyle Y(y)=N(x,y)-{\frac {\partial }{\partial y}}\int ^{x}\mu (\lambda ,y)M(\lambda ,y)\,d\lambda }
13693:
13663:
12488:
12427:
4233:
4218:
1955:
1495:
792:
664:
515:
418:
305:
227:
9895:{\displaystyle 2{\frac {dy}{dx}}{\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\left({\frac {dy}{dx}}\right)^{2}}
13928:
13798:
13759:
13535:
8198:
350:
265:
46:
follows a curve determined by an ordinary differential equation that is derived from Newton's second law.
7596:{\displaystyle {\begin{aligned}P(y){\frac {dy}{dx}}+Q(x)&=0\\P(y)\,dy+Q(x)\,dx&=0\end{aligned}}}
5928:
4364:
1098:
13119:
4907:{\displaystyle I_{\max }=(x_{-},x_{+}),x_{\pm }\in \mathbb {R} \cup \{\pm \infty \},x_{0}\in I_{\max }}
4286:
4272:
555:
423:
13764:
11595:
10869:
14049:
13968:
12283:
4918:
4186:
1572:
526:
504:
14104:
14074:
4162:. Hence, analysts began to substitute the study of functions, thus opening a new and fertile field.
3450:
any ODE of order greater than one can be (and usually is) rewritten as system of ODEs of first order
669:
14133:
13963:
13477:
12229:
11656:
10930:
5780:
4147:
3428:{\displaystyle {\frac {\partial \mathbf {F} (x,\mathbf {u} ,\mathbf {v} )}{\partial \mathbf {v} }}}
520:
428:
5679:
5289:
5182:
4374:
systems, which may have multiple solutions stemming from their (non-linear) algebraic part alone.
4097:{\displaystyle y(x)={\frac {1}{4}}\left(1-{\frac {x}{2}}+\left|1-{\frac {x}{2}}\right|\right)^{2}}
13973:
13943:
13933:
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13384:
12375:
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5603:
This shows clearly that the maximum interval may depend on the initial conditions. The domain of
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4232:
A general solution approach uses the symmetry property of differential equations, the continuous
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379:
10476:{\displaystyle {\frac {d^{2}y}{dx^{2}}}+2p(x){\frac {dy}{dx}}+\left(p(x)^{2}+p'(x)\right)y=q(x)}
1267:
Many mathematicians have studied differential equations and contributed to the field, including
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11714:
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4222:
4217:, be referred to a common source, and that ordinary differential equations that admit the same
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1021:
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13601:
9161:{\displaystyle {\frac {\partial (\mu M)}{\partial y}}={\frac {\partial (\mu N)}{\partial x}}}
4306:
4249:
3860:
754:
659:
644:
533:
476:
458:
297:
53:
12613:. Numerical Analysis 2000. Vol. VI: Ordinary Differential Equations and Integral Equations.
8830:{\displaystyle X(x)=M(x,y)-{\frac {\partial }{\partial x}}\int ^{y}N(x,\lambda )\,d\lambda }
8727:{\displaystyle Y(y)=N(x,y)-{\frac {\partial }{\partial y}}\int ^{x}M(\lambda ,y)\,d\lambda }
4257:
14331:
14236:
13983:
13907:
13876:
13420:
13377:
13182:
13131:
12700:
12618:
11589:
10863:
4350:
4155:
1488:
1229:
545:
481:
454:
13260:
7370:{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=F(y)\\dy&=F(y)\,dx\end{aligned}}}
7199:{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=F(x)\\dy&=F(x)\,dx\end{aligned}}}
1061:
761:. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the
8:
14326:
14316:
14269:
13993:
13886:
13780:
13396:
13297:
Computer
Methods for Ordinary Differential Equations and Differential-Algebraic Equations
12498:
4382:
The theorem can be stated simply as follows. For the equation and initial value problem:
4334:
4127:
1672:
is often used in physics for representing derivatives of low order with respect to time.
1257:
1171:
1159:
577:
562:
463:
327:
166:
133:
124:
14125:
13400:
12622:
12179:{\displaystyle C_{j}e^{\alpha _{j}x}=C_{j}e^{\chi _{j}x}\cos(\gamma _{j}x+\varphi _{j})}
4146:
The primitive attempt in dealing with differential equations had in view a reduction to
3448:
and so forth are also assumed non-singular according to this scheme, although note that
14348:
14231:
14114:
13953:
13948:
13871:
13703:
12955:
12680:
12585:
12065:
9072:
3359:{\displaystyle \mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} '\right)={\boldsymbol {0}}}
1986:
1968:
1469:
1444:) appears on both sides of the differential equation, and is indicated in the notation
1175:
572:
567:
450:
12631:
12606:
6007:
14321:
14284:
13575:
13506:
13485:
13471:
13452:
13444:
13434:
13406:
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13326:
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13282:
13246:
13226:
13214:
13204:
13186:
13175:
13157:
13135:
13088:
13051:
12875:
12659:
12636:
12524:
12371:
9062:{\displaystyle {\frac {\partial M}{\partial y}}\neq {\frac {\partial N}{\partial x}}}
4624:
4620:
4118:
3868:
1413:
1272:
1163:
796:
629:
408:
143:
13553:
12959:
5283:
and therefore locally Lipschitz continuous, satisfying the Picard–Lindelöf theorem.
4289:. The problems are identified as Sturm–Liouville problems (SLP) and are named after
684:
14274:
14254:
14099:
14079:
13557:
13355:
13242:
12947:
12626:
12409:
12385:
4305:
There are several theorems that establish existence and uniqueness of solutions to
4294:
4194:
4176:
2186:
1276:
1179:
694:
679:
12319:
since this is a very common solution that physically behaves in a sinusoidal way.
14289:
14226:
14221:
14216:
14059:
13988:
13498:
13426:
13416:
13373:
13345:
13321:
8418:{\displaystyle {\frac {\partial M}{\partial y}}={\frac {\partial N}{\partial x}}}
7685:{\displaystyle \int ^{y}P(\lambda )\,d\lambda +\int ^{x}Q(\lambda )\,d\lambda =C}
5922:
first-order differential equations by defining a new family of unknown functions
3436:
3367:
1214:
634:
550:
77:
14064:
966:{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}+b(x)=0,}
689:
14054:
13861:
13341:
13170:
13123:
12939:
12564:
6601:
4221:
present comparable integration difficulties. He also emphasized the subject of
4190:
4159:
3581:
3456:
3445:
1186:
654:
639:
445:
433:
152:
27:
Differential equation containing derivatives with respect to only one variable
14362:
14293:
14249:
13579:
13467:
13153:
12640:
4282:
4182:
4135:
2419:
1288:
1190:
13563:
13550:, containing a list of ordinary differential equations with their solutions.
13359:
12951:
8167:{\displaystyle \ln(Cx)=\int ^{xy}{\frac {N(\lambda )\,d\lambda }{\lambda }}}
7876:{\displaystyle \ln(Cx)=\int ^{y/x}{\frac {d\lambda }{F(\lambda )-\lambda }}}
5286:
Even in such a simple setting, the maximum domain of solution cannot be all
1402:{\displaystyle m{\frac {\mathrm {d} ^{2}x(t)}{\mathrm {d} t^{2}}}=F(x(t))\,}
14109:
14044:
13958:
13572:
from the Holistic Numerical Methods Institute, University of South Florida.
13275:
Applications of Lie's Theory of Ordinary and Partial Differential Equations
12607:"Numerical methods for ordinary differential equations in the 20th century"
12556:
4245:
4241:
1268:
674:
624:
510:
138:
13087:
Further Elementary Analysis, R. Porter, G.Bell & Sons (London), 1978,
5996:-dimensional system of first-order coupled differential equations is then
4193:(1873) attacked the theory along lines parallel to those in his theory of
2247:
A number of coupled differential equations form a system of equations. If
13866:
13606:
13333:
13222:
12418:, a proprietary application primarily intended for symbolic calculations.
12415:
4598:
4278:
2962:
1264:(stock trends, interest rates and the market equilibrium price changes).
1241:
1206:
742:
82:
5123:{\displaystyle \lim _{x\to x_{\pm }}y(x)\ \in \partial {\bar {\Omega }}}
3455:
The behavior of a system of ODEs can be visualized through the use of a
14179:
14094:
12400:, a high-level language, primarily intended for numerical computations.
12397:
11710:
6556:
5771:
which is one of the two possible cases according to the above theorem.
4370:
Also, uniqueness theorems like the Lipschitz one above do not apply to
4237:
4210:
762:
699:
39:
35:
30:
12430:, a high-level language primarily intended for numerical computations.
12068:, allows some terms in the previous results to be written in the form
5908:{\displaystyle F\left(x,y,y',y'',\ \ldots ,\ y^{(n-1)}\right)=y^{(n)}}
4753:
to be linear, this applies to non-linear equations that take the form
13856:
13547:
13449:
CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3
13390:
6608:
4214:
1261:
1245:
1237:
440:
161:
104:
94:
12218:
4377:
4309:
involving ODEs both locally and globally. The two main theorems are
4158:(1799) showed, however, that complex differential equations require
3886:
section in this article, and is frequently used when discussing the
14336:
14174:
14169:
13896:
13588:
An introductory textbook on differential equations by Jiri Lebl of
12685:
12590:
12557:"What is the origin of the term "ordinary differential equations"?"
12520:
A First Course in Differential Equations with Modeling Applications
12433:
4768:
4253:
1225:
1218:
1185:
Some ODEs can be solved explicitly in terms of known functions and
800:
10831:{\displaystyle {\frac {d^{2}y}{dx^{2}}}+b{\frac {dy}{dx}}+cy=r(x)}
4921:
of the solution that satisfies this initial condition with domain
4917:
such that any solution that satisfies this initial condition is a
4790:) there exists a unique maximum (possibly infinite) open interval
12445:
11557:{\displaystyle \sum _{j=0}^{n}b_{j}{\frac {d^{j}y}{dx^{j}}}=r(x)}
1280:
1253:
1249:
1233:
1205:
Ordinary differential equations (ODEs) arise in many contexts of
1167:
765:
of those functions. The term "ordinary" is used in contrast with
115:
110:
99:
13585:
13230:
4185:
inspired a novel approach, subsequently elaborated by Thomé and
13350:, Classics in Applied Mathematics, vol. 38, Philadelphia:
13203:(2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003.
13201:
Handbook of Exact Solutions for Ordinary Differential Equations
12449:
12403:
12391:
12381:
4167:
variables, and, if so, what are the characteristic properties.
4163:
1221:
of quantities, which is how they enter differential equations.
1210:
776:
one independent variable, and, less commonly, in contrast with
43:
1920:{\displaystyle F\left(x,y,y',y'',\ \ldots ,\ y^{(n)}\right)=0}
1798:{\displaystyle F\left(x,y,y',\ldots ,y^{(n-1)}\right)=y^{(n)}}
13219:
Differential Equations with Applications and Historical Notes
12455:
12439:
12421:
12323:
together to obtain the general solution to the ODE, that is:
7771:{\displaystyle {\frac {dy}{dx}}=F\left({\frac {y}{x}}\right)}
7447:{\displaystyle x=\int ^{y}{\frac {d\lambda }{F(\lambda )}}+C}
2103:{\displaystyle y^{(n)}=\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x)}
1292:
12388:) software package for the integration and analysis of ODEs.
3692:{\displaystyle F(x,u,u',\ \ldots ,\ u^{(n)})=0\quad x\in I.}
1665:{\displaystyle ({\dot {y}},{\ddot {y}},{\overset {...}{y}})}
1189:. When that is not possible, the equation for computing the
13589:
12484:
List of dynamical systems and differential equations topics
12442:, a Python package that includes an ODE integration module.
10736:
Second-order, linear, inhomogeneous, constant coefficients
10725:{\displaystyle y=e^{-\int ^{x}P(\lambda )\,d\lambda }\left}
10323:
Second-order, linear, inhomogeneous, function coefficients
10311:{\displaystyle y=e^{-\int ^{x}P(\lambda )\,d\lambda }\left}
5586:
5048:{\displaystyle \limsup _{x\to x_{\pm }}\|y(x)\|\to \infty }
3987:{\displaystyle y'=-{\text{sgn}}(y){\sqrt {|y|}},\,\,y(0)=1}
13602:
Solving an ordinary differential equation in Wolfram|Alpha
11869:{\displaystyle y_{c}=\sum _{j=1}^{n}C_{j}e^{\alpha _{j}x}}
10042:
First-order, linear, inhomogeneous, function coefficients
6259:{\displaystyle \mathbf {y} '=\mathbf {F} (x,\mathbf {y} )}
5779:
Differential equations are usually easier to solve if the
1302:
of motion—the relationship between the displacement
14155:
13570:
A primer on analytical solution of differential equations
13386:
A Treatise on Ordinary and Partial Differential Equations
5761:{\displaystyle \lim _{x\to x_{\pm }}\|y(x)\|\to \infty ,}
11253:{\displaystyle y_{c}=(C_{1}x+C_{2})e^{-{\frac {bx}{2}}}}
4300:
799:
in the unknown function and its derivatives, that is an
13586:
Notes on Diffy Qs: Differential Equations for Engineers
12869:
12406:, an open source application for numerical computation.
12394:, a technical computing application (MATrix LABoratory)
12342:
general solution of the associated homogeneous equation
11466:
th-order, linear, inhomogeneous, constant coefficients
4765:), and it can also be applied to systems of equations.
1147:
are the successive derivatives of the unknown function
13503:
Handbook of First Order Partial Differential Equations
13239:
Physics for Scientists and Engineers: Extended version
12583:
12412:, a proprietary application for symbolic calculations.
11726:
5665:{\displaystyle \mathbb {R} \setminus (x_{0}+1/y_{0}),}
5390:{\displaystyle y(x)={\frac {y_{0}}{(x_{0}-x)y_{0}+1}}}
4260:, and finally finding exact analytic solutions to DE.
3544:{\displaystyle F\left(x,y,y',\ldots ,y^{(n)}\right)=0}
3449:
3254:
2980:
2534:
2437:
1193:
of the solutions may be useful. For applied problems,
13576:
Ordinary Differential Equations and Dynamical Systems
13473:
Ordinary Differential Equations and Dynamical Systems
12331:
12286:
12074:
11898:
11801:
11659:
11598:
11473:
11280:
11179:
10992:
10933:
10872:
10743:
10546:
10492:
10330:
10181:
10122:
10049:
9912:
9781:
9700:
9553:
9432:
9193:
9096:
9015:
8852:
8740:
8637:
8436:
8371:
8208:
8066:
7894:
7801:
7723:
7614:
7472:
7395:
7283:
7217:
7112:
6952:
6692:
6278:
6220:
6005:
5931:
5804:
5707:
5682:
5613:
5409:
5317:
5292:
5232:
5185:
5143:
5063:
4997:
4957:
4927:
4799:
4633:
4483:
4388:
4006:
3915:
3769:
3609:
3476:
3375:
3308:
2974:
2856:
2431:
2280:
2003:
1971:
1835:
1711:
1607:
1323:
1195:
numerical methods for ordinary differential equations
1101:
1064:
1024:
984:
812:
12452:, for computing with functions to 15-digit accuracy.
11779:{\textstyle \prod _{j=1}^{n}(\alpha -\alpha _{j})=0}
10165:{\displaystyle e^{\int ^{x}P(\lambda )\,d\lambda }.}
10019:
5786:
2198:
1174:
are solutions of linear differential equations (see
12798:
12796:
12479:
Laplace transform applied to differential equations
12424:, a Python package that can solve ODEs symbolically
10532:{\displaystyle e^{\int ^{x}P(\lambda )\,d\lambda }}
7262:{\displaystyle y=\int ^{x}F(\lambda )\,d\lambda +C}
13325:
13319:
13174:
12759:
12757:
12355:
12311:
12178:
12025:
11868:
11778:
11691:
11621:
11588:substitute and solve polynomial in α, to find the
11556:
11450:
11252:
11151:
10965:
10895:
10862:substitute and solve polynomial in α, to find the
10830:
10724:
10531:
10475:
10310:
10164:
10105:
10006:
9894:
9752:
9660:
9539:
9416:
9160:
9061:
8999:
8829:
8726:
8622:
8417:
8355:
8166:
8042:
7875:
7770:
7684:
7595:
7446:
7369:
7261:
7198:
7090:
6920:
6462:
6258:
6200:
5973:
5907:
5760:
5690:
5664:
5592:
5389:
5300:
5256:
5193:
5161:
5122:
5047:
4979:
4940:
4906:
4727:
4577:
4457:
4096:
3986:
3813:
3691:
3543:
3427:
3358:
3291:
2946:
2832:
2402:explicit system of ordinary differential equations
2389:
2102:
1977:
1919:
1797:
1664:
1401:
1137:
1079:
1046:
1006:
965:
14024:List of nonlinear ordinary differential equations
13391:University of Michigan Historical Math Collection
12944:1985 24th IEEE Conference on Decision and Control
9674:
5970:
4378:Local existence and uniqueness theorem simplified
14360:
14029:List of nonlinear partial differential equations
13517:Handbook of Differential Equations (3rd edition)
13003:Logan, J. (2013). Applied mathematics (4th ed.).
12863:
12793:
12611:Journal of Computational and Applied Mathematics
7697:
6655:, without adding constants (explicitly stated).
5709:
5065:
4999:
4933:
4899:
4805:
4769:Global uniqueness and maximum domain of solution
4458:{\displaystyle y'=F(x,y)\,,\quad y_{0}=y(x_{0})}
795:is a differential equation that is defined by a
13118:
12989:harvtxt error: no target: CITEREFLawrence1999 (
12937:
12754:
12736:
11451:{\displaystyle y_{c}=e^{-{\frac {bx}{2}}}\left}
4201:= 0 under rational one-to-one transformations.
3897:
14019:List of linear ordinary differential equations
13352:Society for Industrial and Applied Mathematics
13258:
12872:Surveys in Differential-Algebraic Equations II
12516:
3874:In the context of linear ODE, the terminology
14141:
13622:
13294:
13017:harvtxt error: no target: CITEREFAscher1998 (
12899:harvtxt error: no target: CITEREFAscher1998 (
12855:harvtxt error: no target: CITEREFAscher1998 (
9753:{\displaystyle {\frac {d^{2}y}{dx^{2}}}=F(y)}
6473:
5172:Note that the maximum domain of the solution
4285:of linear operators defined via second-order
3824:A solution that has no extension is called a
1197:can supply an approximation of the solution.
722:
13548:EqWorld: The World of Mathematical Equations
6685:(general case, see below for special cases)
5791:Any explicit differential equation of order
5746:
5731:
5036:
5021:
4875:
4866:
757:(DE) dependent on only a single independent
13431:Lectures on Ordinary Differential Equations
12365:
12247:. Unsourced material may be challenged and
10106:{\displaystyle {\frac {dy}{dx}}+P(x)y=Q(x)}
7785:, then solve by separation of variables in
4141:
3847:th-order equation is a solution containing
2232:A differential equation that is not linear.
14148:
14134:
13636:
13629:
13615:
5176:is always an interval (to have uniqueness)
2251:is a vector whose elements are functions;
1252:(infectious diseases, genetic variation),
729:
715:
13443:
13328:Theory of Ordinary Differential Equations
13046:
13044:
13042:
13040:
13038:
13036:
13034:
13032:
13030:
13028:
12684:
12653:
12630:
12589:
12267:Learn how and when to remove this message
11409:
11348:
11107:
11034:
10670:
10649:
10583:
10520:
10290:
10269:
10218:
10150:
9968:
9651:
9530:
9397:
9365:
9301:
9269:
8976:
8948:
8820:
8717:
8603:
8571:
8526:
8494:
8332:
8304:
8116:
8019:
7991:
7938:
7669:
7637:
7572:
7550:
7356:
7246:
7185:
7075:
7009:
6897:
6849:
6776:
5684:
5615:
5419:
5294:
5187:
5162:{\displaystyle \partial {\bar {\Omega }}}
4859:
4418:
4266:
3965:
3964:
3810:
1813:ordinary differential equation of order n
1398:
786:
13169:
12984:
12925:
12913:
12838:
12802:
12775:
12712:
12195:is an arbitrary constant (phase shift).
4980:{\displaystyle x_{\pm }\neq \pm \infty }
3883:
3814:{\displaystyle u(x)=v(x)\quad x\in I.\,}
1822:ordinary differential equation of order
1314:, is given by the differential equation
783:(SDEs) where the progression is random.
29:
13340:
13272:
13213:
13102:
13100:
13074:
13072:
12814:
12787:
12724:
12604:
12510:
1571:is more useful for differentiation and
14:
14361:
13505:, Taylor & Francis, London, 2002.
13466:
13259:Boscain, Ugo; Chitour, Yacine (2011),
13236:
13147:
13025:
13012:
12894:
12850:
12826:
12763:
12748:
12699:Mathematics for Chemists, D.M. Hirst,
12202:
7273:First-order, autonomous, separable in
6658:
5203:may depend on the specific choice of (
4987:, there are exactly two possibilities
2843:These are not necessarily linear. The
1945:if it does not depend on the variable
14129:
13610:
13554:Online Notes / Differential Equations
12678:
12448:, an open-source package, written in
9186:can be found in a suitable way, then
5774:
4477:are continuous in a closed rectangle
4301:Existence and uniqueness of solutions
4112:
3997:Admits the finite duration solution:
3366:, some sources also require that the
2185:. In this case, there is always the "
1675:
1224:Specific mathematical fields include
18:Non-homogeneous differential equation
14014:List of named differential equations
13395:
13295:Ascher, Uri; Petzold, Linda (1998),
13150:Introduction to Mathematical Physics
13097:
13069:
12940:"Finite Time Differential Equations"
12537:from the original on 17 January 2020
12245:adding citations to reliable sources
12212:
2159:, leading to further classification.
1995:; that is, it can be rewritten as
1232:. Scientific fields include much of
1093:that do not need to be linear, and
772:(PDEs) which may be with respect to
177:List of named differential equations
13939:Method of undetermined coefficients
13720:Dependent and independent variables
13501:, V. F. Zaitsev, and A. Moussiaux,
13199:Polyanin, A. D. and V. F. Zaitsev,
12494:Method of undetermined coefficients
12209:Method of undetermined coefficients
9763:Multiply both sides of equation by
8053:Separation of variables (divide by
6931:Separation of variables (divide by
6211:more compactly in vector notation:
3888:method of undetermined coefficients
2206:A linear differential equation is
1930:There are further classifications:
250:Dependent and independent variables
24:
12870:Achim Ilchmann; Timo Reis (2014).
12703:, 1976, (No ISBN) SBN: 333-18172-7
12474:Examples of differential equations
9596:
9592:
9475:
9471:
9149:
9132:
9117:
9100:
9050:
9042:
9027:
9019:
8783:
8779:
8680:
8676:
8406:
8398:
8383:
8375:
5974:{\displaystyle y_{i}=y^{(i-1)}.\!}
5752:
5557:
5453:
5150:
5144:
5111:
5105:
5042:
4974:
4872:
4170:
3414:
3379:
2170:A linear differential equation is
1357:
1332:
1166:functions that are encountered in
1138:{\displaystyle y',\ldots ,y^{(n)}}
25:
14385:
13531:"Differential equation, ordinary"
13523:
13066:Boscain; Chitour 2011, p. 21
11640:method of variation of parameters
10914:method of variation of parameters
5787:Reduction to a first-order system
5619:
5133:where Ω is the open set in which
2236:
2201:Nonhomogeneous (or inhomogeneous)
1472:representing an unknown function
13836:Carathéodory's existence theorem
13405:, Dover Publications, New York,
13389:, John Wiley and Sons, 1913, in
13177:Advanced Engineering Mathematics
12517:Dennis G. Zill (15 March 2012).
12217:
11622:{\displaystyle e^{\alpha _{j}x}}
10896:{\displaystyle e^{\alpha _{j}x}}
6327:
6280:
6249:
6235:
6223:
5783:of the equation can be reduced.
3442:differential algebraic equations
3418:
3405:
3397:
3383:
3352:
3335:
3326:
3310:
3219:
3200:
3187:
3178:
3128:
3109:
3096:
3087:
3044:
3025:
3012:
3003:
2940:
2915:
2896:
2883:
2874:
2858:
2795:
2776:
2763:
2754:
2694:
2675:
2662:
2653:
2600:
2581:
2568:
2559:
2360:
2341:
2328:
2319:
2303:
2283:
2243:System of differential equations
1594:is more useful for representing
385:Carathéodory's existence theorem
292:
14374:Ordinary differential equations
14280:Least-squares spectral analysis
14207:Fundamental theorem of calculus
13596:Modeling with ODEs using Scilab
13519:, Academic Press, Boston, 1997.
13402:Ordinary Differential Equations
13347:Ordinary differential equations
13313:
13279:Institute of Physics Publishing
13081:
13060:
13006:
12997:
12978:
12966:
12931:
12919:
12907:
12888:
12844:
12832:
12820:
12808:
12781:
12769:
12742:
12730:
12656:Ordinary differential equations
12312:{\displaystyle y=Ae^{\alpha t}}
6325:
4422:
4355:local existence and uniqueness
4204:
3828:. A solution defined on all of
3797:
3676:
1702:. Then an equation of the form
12874:. Springer. pp. 104–105.
12718:
12706:
12693:
12672:
12654:Greenberg, Michael D. (2012).
12647:
12598:
12577:
12549:
12173:
12144:
11767:
11748:
11551:
11545:
11222:
11193:
10825:
10819:
10667:
10661:
10646:
10640:
10580:
10574:
10517:
10511:
10470:
10464:
10447:
10441:
10421:
10414:
10380:
10374:
10287:
10281:
10266:
10260:
10215:
10209:
10147:
10141:
10100:
10094:
10082:
10076:
9965:
9959:
9747:
9741:
9675:General second-order equations
9648:
9636:
9630:
9618:
9584:
9572:
9563:
9557:
9527:
9515:
9509:
9497:
9463:
9451:
9442:
9436:
9394:
9388:
9362:
9350:
9344:
9332:
9298:
9292:
9266:
9254:
9248:
9236:
9213:
9201:
9144:
9135:
9112:
9103:
8973:
8961:
8945:
8933:
8913:
8901:
8872:
8860:
8817:
8805:
8771:
8759:
8750:
8744:
8714:
8702:
8668:
8656:
8647:
8641:
8600:
8594:
8568:
8556:
8523:
8517:
8491:
8479:
8456:
8444:
8329:
8317:
8301:
8289:
8269:
8257:
8228:
8216:
8158:
8155:
8149:
8140:
8134:
8128:
8113:
8107:
8082:
8073:
8016:
8007:
7988:
7979:
7935:
7926:
7914:
7905:
7861:
7855:
7817:
7808:
7666:
7660:
7634:
7628:
7569:
7563:
7547:
7541:
7521:
7515:
7486:
7480:
7432:
7426:
7353:
7347:
7324:
7318:
7243:
7237:
7182:
7176:
7153:
7147:
7069:
7063:
7048:
7042:
7003:
6997:
6982:
6976:
6894:
6888:
6875:
6869:
6846:
6840:
6827:
6821:
6773:
6767:
6754:
6748:
6732:
6726:
6713:
6707:
6454:
6451:
6413:
6375:
6369:
6331:
6319:
6287:
6253:
6239:
6188:
6150:
5962:
5950:
5918:can be written as a system of
5900:
5894:
5876:
5864:
5749:
5743:
5737:
5716:
5656:
5622:
5365:
5346:
5327:
5321:
5153:
5114:
5096:
5090:
5072:
5039:
5033:
5027:
5006:
4839:
4813:
4722:
4684:
4678:
4640:
4572:
4534:
4528:
4490:
4452:
4439:
4415:
4403:
4123:partial differential equations
4016:
4010:
3975:
3969:
3955:
3947:
3941:
3935:
3794:
3788:
3779:
3773:
3667:
3662:
3656:
3613:
3525:
3519:
3467:Given a differential equation
3409:
3387:
3235:
3230:
3224:
3168:
3144:
3139:
3133:
3077:
3060:
3055:
3049:
2993:
2926:
2920:
2812:
2800:
2711:
2699:
2617:
2605:
2513:
2507:
2481:
2475:
2456:
2450:
2377:
2365:
2294:
2288:
2162:
2097:
2091:
2080:
2074:
2066:
2060:
2015:
2009:
1901:
1895:
1790:
1784:
1766:
1754:
1682:Order of differential equation
1659:
1608:
1459:
1424:is a function of the position
1395:
1392:
1386:
1380:
1351:
1345:
1130:
1124:
1074:
1068:
1041:
1035:
1001:
995:
951:
945:
934:
928:
920:
914:
884:
878:
854:
848:
829:
823:
747:ordinary differential equation
472: / Integral solutions
13:
1:
13482:American Mathematical Society
13112:
12632:10.1016/S0377-0427(00)00455-6
12605:Butcher, J. C. (2000-12-15).
11692:{\displaystyle y=y_{c}+y_{p}}
10966:{\displaystyle y=y_{c}+y_{p}}
7698:General first-order equations
5057:leaves domain of definition:
4291:J. C. F. Sturm
4234:infinitesimal transformations
4219:infinitesimal transformations
2224:
1933:
1310:of an object under the force
1200:
13664:Notation for differentiation
13277:, Bristol and Philadelphia:
13262:Introduction à l'automatique
12737:Halliday & Resnick (1977
12489:Matrix differential equation
12428:Julia (programming language)
5691:{\displaystyle \mathbb {R} }
5301:{\displaystyle \mathbb {R} }
5194:{\displaystyle \mathbb {R} }
4776:For each initial condition (
4627:, then there is an interval
4287:homogeneous linear equations
3898:Solutions of finite duration
3576:is an interval, is called a
3462:
2143:are continuous functions of
1496:notation for differentiation
793:linear differential equation
516:Exponential response formula
262:Coupled / Decoupled
7:
13760:Exact differential equation
13536:Encyclopedia of Mathematics
12462:
6645:the integration substitute
4236:of solutions to solutions (
4107:
1961:A differential equation is
1941:A differential equation is
10:
14390:
13273:Dresner, Lawrence (1999),
13241:(3rd ed.), New York:
13181:(3rd ed.), New York:
13130:(3rd ed.), New York:
12206:
7458:First-order, separable in
7102:First-order, separable in
6677:First-order, separable in
6583:are real given constants;
6474:Summary of exact solutions
5676:The maximum domain is not
5400:which has maximum domain:
4991:explosion in finite time:
4728:{\displaystyle I=\subset }
4270:
4223:transformations of contact
4174:
3906:As example, the equation:
2271:and its derivatives, then
2240:
1679:
1432:) of the particle at time
1260:(population competition),
14345:
14245:
14164:
14070:Józef Maria Hoene-Wroński
14050:Gottfried Wilhelm Leibniz
14037:
14006:
13916:
13849:
13841:Cauchy–Kowalevski theorem
13818:
13811:
13773:
13712:
13651:
13644:
13451:. Providence: CRC-Press.
11709:are the solutions of the
11642:, though for very simple
10916:, though for very simple
10036:
10033:
10030:
9693:Second-order, autonomous
9687:
9684:
9681:
7716:First-order, homogeneous
7710:
7707:
7704:
6671:
6668:
6665:
6600:are arbitrary constants (
4941:{\displaystyle I_{\max }}
4623:, square brackets denote
4578:{\displaystyle R=\times }
3596:-times differentiable on
3302:For a system of the form
1953:
650:Józef Maria Hoene-Wroński
596:Undetermined coefficients
505:Method of characteristics
390:Cauchy–Kowalevski theorem
13964:Finite difference method
13237:Tipler, Paul A. (1991),
13148:Harper, Charlie (1976),
12938:Vardia T. Haimo (1985).
12504:
12366:Software for ODE solving
6626:just means to integrate
5607:could be taken as being
5257:{\displaystyle y'=y^{2}}
4142:Reduction to quadratures
3853:constants of integration
2961:= (0, 0, ..., 0) is the
1596:higher-order derivatives
1091:differentiable functions
1047:{\displaystyle a_{n}(x)}
1007:{\displaystyle a_{0}(x)}
375:Picard–Lindelöf theorem
369:Existence and uniqueness
13944:Variation of parameters
13934:Separation of variables
13831:Peano existence theorem
13826:Picard–Lindelöf theorem
13713:Attributes of variables
13360:10.1137/1.9780898719222
12952:10.1109/CDC.1985.268832
12658:. Hoboken, N.J: Wiley.
12376:computer algebra system
11650:) inspection may work.
11567:Complementary function
10924:) inspection may work.
10841:Complementary function
7887:First-order, separable
7381:Separation of variables
4345:Picard–Lindelöf theorem
4329:Peano existence theorem
4256:, recursion operators,
3892:variation of parameters
1436:. The unknown function
1240:(celestial mechanics),
601:Variation of parameters
591:Separation of variables
380:Peano existence theorem
14212:Calculus of variations
14185:Differential equations
14105:Carl David Tolmé Runge
13679:Differential-algebraic
13638:Differential equations
13564:Differential Equations
13433:, Dover Publications,
12946:. pp. 1729–1733.
12469:Boundary value problem
12357:
12313:
12180:
12042:complex, then setting
12027:
11965:
11932:
11870:
11835:
11780:
11747:
11693:
11623:
11558:
11494:
11452:
11254:
11153:
10967:
10897:
10832:
10726:
10533:
10477:
10312:
10166:
10107:
10031:Differential equation
10008:
9904:then integrate twice.
9896:
9754:
9682:Differential equation
9662:
9541:
9418:
9162:
9063:
9001:
8831:
8728:
8624:
8428:Integrate throughout.
8419:
8357:
8168:
8044:
7877:
7772:
7705:Differential equation
7686:
7606:Integrate throughout.
7597:
7448:
7371:
7263:
7200:
7092:
6922:
6666:Differential equation
6464:
6260:
6202:
5975:
5909:
5762:
5692:
5666:
5594:
5391:
5308:since the solution is
5302:
5258:
5195:
5163:
5124:
5049:
4981:
4942:
4908:
4729:
4579:
4459:
4307:initial value problems
4273:Sturm–Liouville theory
4267:Sturm–Liouville theory
4098:
3988:
3851:arbitrary independent
3815:
3693:
3545:
3429:
3360:
3293:
2948:
2834:
2391:
2265:vector-valued function
2104:
2049:
1989:of the derivatives of
1979:
1921:
1799:
1666:
1403:
1139:
1081:
1048:
1008:
967:
787:Differential equations
781:differential equations
770:differential equations
670:Carl David Tolmé Runge
213:Differential-algebraic
54:Differential equations
47:
14369:Differential calculus
14305:Representation theory
14264:quaternionic analysis
14260:Hypercomplex analysis
14158:mathematical analysis
14090:Augustin-Louis Cauchy
14085:Joseph-Louis Lagrange
13979:Finite element method
13969:Crank–Nicolson method
13903:Numerical integration
13882:Exponential stability
13774:Relation to processes
13659:Differential operator
13566:, S.O.S. Mathematics.
13320:Coddington, Earl A.;
12561:hsm.stackexchange.com
12358:
12314:
12181:
12028:
11938:
11912:
11871:
11815:
11781:
11727:
11694:
11624:
11559:
11474:
11453:
11255:
11154:
10968:
10898:
10833:
10727:
10534:
10478:
10313:
10167:
10108:
10009:
9897:
9755:
9663:
9542:
9419:
9163:
9064:
9002:
8832:
8729:
8625:
8420:
8358:
8169:
8045:
7878:
7773:
7687:
7598:
7449:
7372:
7264:
7201:
7093:
6923:
6465:
6261:
6203:
5976:
5910:
5763:
5693:
5667:
5595:
5392:
5303:
5259:
5196:
5164:
5125:
5050:
4982:
4943:
4909:
4730:
4580:
4460:
4365:Grönwall's inequality
4339:local existence only
4250:differential geometry
4099:
3989:
3816:
3694:
3546:
3430:
3361:
3294:
2949:
2835:
2392:
2105:
2023:
1980:
1922:
1800:
1698:, and derivatives of
1667:
1412:which constrains the
1404:
1140:
1082:
1049:
1009:
968:
755:differential equation
660:Augustin-Louis Cauchy
645:Joseph-Louis Lagrange
477:Numerical integration
459:Exponential stability
322:Relation to processes
33:
14237:Table of derivatives
13984:Finite volume method
13908:Dirac delta function
13877:Asymptotic stability
13819:Existence/uniqueness
13684:Integro-differential
12523:. Cengage Learning.
12386:Artistic License 2.0
12329:
12284:
12241:improve this section
12072:
11896:
11799:
11724:
11657:
11631:Particular integral
11596:
11590:linearly independent
11471:
11278:
11177:
10990:
10931:
10905:Particular integral
10870:
10864:linearly independent
10741:
10544:
10490:
10486:Integrating factor:
10328:
10179:
10120:
10116:Integrating factor:
10047:
9910:
9779:
9698:
9551:
9430:
9191:
9094:
9013:
8850:
8843:Inexact differential
8738:
8635:
8434:
8369:
8206:
8064:
7892:
7799:
7721:
7612:
7470:
7393:
7281:
7215:
7209:Direct integration.
7110:
6950:
6690:
6611:), and the notation
6481:In the table below,
6276:
6218:
6003:
5929:
5802:
5705:
5680:
5611:
5407:
5315:
5290:
5230:
5183:
5179:may be smaller than
5141:
5061:
4995:
4955:
4925:
4797:
4631:
4481:
4386:
4351:Lipschitz continuous
4004:
3913:
3767:
3702:Given two solutions
3607:
3474:
3373:
3306:
2972:
2854:
2429:
2278:
2001:
1985:can be written as a
1969:
1833:
1709:
1605:
1489:independent variable
1414:motion of a particle
1321:
1298:A simple example is
1244:(weather modeling),
1230:analytical mechanics
1099:
1080:{\displaystyle b(x)}
1062:
1022:
982:
810:
482:Dirac delta function
218:Integro-differential
14317:Continuous function
14270:Functional analysis
13994:Perturbation theory
13974:Runge–Kutta methods
13954:Integral transforms
13887:Rate of convergence
13783:(discrete analogue)
12623:2000JCoAM.125....1B
12499:Recurrence relation
12350:particular solution
12203:The guessing method
6659:Separable equations
6139:
6105:
6056:
6022:
3876:particular solution
3865:boundary conditions
3857:particular solution
2517:
2485:
2460:
1818:More generally, an
1577:Lagrange's notation
1300:Newton's second law
1258:population modeling
1172:applied mathematics
578:Perturbation theory
573:Integral transforms
464:Rate of convergence
330:(discrete analogue)
167:Population dynamics
134:Continuum mechanics
125:Applied mathematics
14349:Mathematics portal
14232:Lists of integrals
14115:Sofya Kovalevskaya
13949:Integrating factor
13872:Lyapunov stability
13792:Stochastic partial
13445:Ibragimov, Nail H.
13215:Simmons, George F.
12353:
12309:
12176:
12023:
11866:
11776:
11689:
11619:
11554:
11448:
11250:
11149:
10963:
10893:
10828:
10722:
10529:
10473:
10308:
10162:
10103:
10024:th order equations
10004:
9892:
9750:
9658:
9537:
9414:
9412:
9158:
9073:Integration factor
9059:
8997:
8995:
8827:
8724:
8620:
8618:
8415:
8353:
8351:
8199:Exact differential
8184:, the solution is
8164:
8040:
8038:
7873:
7768:
7682:
7593:
7591:
7444:
7367:
7365:
7259:
7196:
7194:
7088:
6918:
6916:
6460:
6256:
6198:
6196:
6127:
6087:
6044:
6010:
5971:
5905:
5775:Reduction of order
5758:
5730:
5688:
5662:
5590:
5585:
5387:
5298:
5254:
5191:
5159:
5120:
5086:
5045:
5020:
4977:
4938:
4904:
4725:
4575:
4455:
4281:and corresponding
4258:Bäcklund transform
4119:singular solutions
4113:Singular solutions
4094:
3984:
3861:initial conditions
3811:
3689:
3541:
3425:
3356:
3289:
3283:
3240:
2944:
2830:
2824:
2520:
2497:
2465:
2440:
2387:
2100:
1987:linear combination
1975:
1917:
1795:
1676:General definition
1662:
1500:Leibniz's notation
1470:dependent variable
1464:In what follows,
1399:
1248:(reaction rates),
1176:Holonomic function
1135:
1077:
1044:
1004:
963:
568:Integrating factor
409:Initial conditions
344:Stochastic partial
48:
14356:
14355:
14322:Special functions
14285:Harmonic analysis
14123:
14122:
14002:
14001:
13807:
13806:
13578:lecture notes by
13556:by Paul Dawkins,
13491:978-0-8218-8328-0
13412:978-0-486-60349-0
13369:978-0-89871-510-1
13306:978-1-61197-139-2
12881:978-3-319-11050-9
12751:, pp. 78–83)
12665:978-1-118-23002-2
12530:978-1-285-40110-2
12374:, an open-source
12351:
12343:
12335:
12277:
12276:
12269:
12200:
12199:
11638:: in general the
11537:
11436:
11432:
11375:
11371:
11315:
11246:
11140:
11105:
11067:
11032:
10912:: in general the
10802:
10776:
10401:
10363:
10068:
10037:General solution
10017:
10016:
9989:
9988:
9880:
9855:
9837:
9803:
9733:
9688:General solution
9672:
9671:
9603:
9482:
9156:
9124:
9057:
9034:
8893:
8790:
8687:
8413:
8390:
8249:
8162:
7957:
7871:
7762:
7742:
7711:General solution
7695:
7694:
7507:
7436:
7306:
7135:
7073:
7007:
6795:
6672:General solution
5858:
5849:
5708:
5549:
5487:
5385:
5169:is its boundary.
5156:
5117:
5101:
5064:
4998:
4951:In the case that
4621:Cartesian product
4359:
4358:
4295:J. Liouville
4195:Abelian integrals
4076:
4052:
4030:
3959:
3933:
3869:singular solution
3650:
3641:
3423:
2965:. In matrix form
1978:{\displaystyle F}
1889:
1880:
1657:
1635:
1620:
1600:Newton's notation
1416:of constant mass
1372:
797:linear polynomial
739:
738:
630:Gottfried Leibniz
521:Finite difference
313:
312:
174:
173:
144:Dynamical systems
16:(Redirected from
14381:
14275:Fourier analysis
14255:Complex analysis
14156:Major topics in
14150:
14143:
14136:
14127:
14126:
14100:Phyllis Nicolson
14080:Rudolf Lipschitz
13917:Solution methods
13892:Series solutions
13816:
13815:
13649:
13648:
13631:
13624:
13617:
13608:
13607:
13558:Lamar University
13544:
13495:
13462:
13423:
13380:
13337:
13331:
13322:Levinson, Norman
13309:
13291:
13269:
13267:
13255:
13243:Worth Publishers
13233:
13195:
13180:
13166:
13144:
13107:
13104:
13095:
13085:
13079:
13076:
13067:
13064:
13058:
13048:
13023:
13022:
13010:
13004:
13001:
12995:
12994:
12982:
12976:
12970:
12964:
12963:
12935:
12929:
12923:
12917:
12911:
12905:
12904:
12892:
12886:
12885:
12867:
12861:
12860:
12848:
12842:
12836:
12830:
12824:
12818:
12812:
12806:
12800:
12791:
12785:
12779:
12773:
12767:
12761:
12752:
12746:
12740:
12734:
12728:
12727:, pp. 1, 2)
12722:
12716:
12710:
12704:
12697:
12691:
12690:
12688:
12676:
12670:
12669:
12651:
12645:
12644:
12634:
12602:
12596:
12595:
12593:
12581:
12575:
12574:
12572:
12571:
12553:
12547:
12546:
12544:
12542:
12514:
12362:
12360:
12359:
12354:
12352:
12349:
12344:
12341:
12336:
12334:general solution
12333:
12318:
12316:
12315:
12310:
12308:
12307:
12272:
12265:
12261:
12258:
12252:
12221:
12213:
12185:
12183:
12182:
12177:
12172:
12171:
12156:
12155:
12137:
12136:
12132:
12131:
12117:
12116:
12104:
12103:
12099:
12098:
12084:
12083:
12032:
12030:
12029:
12024:
12022:
12021:
12017:
12016:
12002:
11998:
11997:
11996:
11981:
11980:
11964:
11963:
11962:
11952:
11931:
11926:
11908:
11907:
11875:
11873:
11872:
11867:
11865:
11864:
11860:
11859:
11845:
11844:
11834:
11829:
11811:
11810:
11785:
11783:
11782:
11777:
11766:
11765:
11746:
11741:
11698:
11696:
11695:
11690:
11688:
11687:
11675:
11674:
11628:
11626:
11625:
11620:
11618:
11617:
11613:
11612:
11587:
11563:
11561:
11560:
11555:
11538:
11536:
11535:
11534:
11521:
11517:
11516:
11506:
11504:
11503:
11493:
11488:
11457:
11455:
11454:
11449:
11447:
11443:
11442:
11438:
11437:
11431:
11430:
11412:
11411:
11394:
11393:
11381:
11377:
11376:
11370:
11369:
11351:
11350:
11333:
11332:
11318:
11317:
11316:
11311:
11303:
11290:
11289:
11271:
11259:
11257:
11256:
11251:
11249:
11248:
11247:
11242:
11234:
11221:
11220:
11205:
11204:
11189:
11188:
11170:
11158:
11156:
11155:
11150:
11148:
11147:
11146:
11142:
11141:
11130:
11129:
11120:
11106:
11098:
11088:
11087:
11075:
11074:
11073:
11069:
11068:
11057:
11056:
11047:
11033:
11025:
11015:
11014:
11002:
11001:
10983:
10972:
10970:
10969:
10964:
10962:
10961:
10949:
10948:
10902:
10900:
10899:
10894:
10892:
10891:
10887:
10886:
10861:
10837:
10835:
10834:
10829:
10803:
10801:
10793:
10785:
10777:
10775:
10774:
10773:
10760:
10756:
10755:
10745:
10731:
10729:
10728:
10723:
10721:
10717:
10716:
10715:
10700:
10699:
10681:
10677:
10657:
10656:
10636:
10635:
10621:
10620:
10606:
10605:
10591:
10590:
10570:
10569:
10538:
10536:
10535:
10530:
10528:
10527:
10507:
10506:
10482:
10480:
10479:
10474:
10454:
10450:
10440:
10429:
10428:
10402:
10400:
10392:
10384:
10364:
10362:
10361:
10360:
10347:
10343:
10342:
10332:
10317:
10315:
10314:
10309:
10307:
10303:
10277:
10276:
10256:
10255:
10241:
10240:
10226:
10225:
10205:
10204:
10174:Armour formula:
10171:
10169:
10168:
10163:
10158:
10157:
10137:
10136:
10112:
10110:
10109:
10104:
10069:
10067:
10059:
10051:
10034:Solution method
10028:
10027:
10013:
10011:
10010:
10005:
10003:
10002:
9990:
9987:
9986:
9955:
9954:
9942:
9941:
9933:
9931:
9930:
9903:
9901:
9899:
9898:
9893:
9891:
9890:
9885:
9881:
9879:
9871:
9863:
9856:
9854:
9843:
9838:
9836:
9835:
9834:
9821:
9817:
9816:
9806:
9804:
9802:
9794:
9786:
9773:
9759:
9757:
9756:
9751:
9734:
9732:
9731:
9730:
9717:
9713:
9712:
9702:
9685:Solution method
9679:
9678:
9667:
9665:
9664:
9659:
9614:
9613:
9604:
9602:
9591:
9546:
9544:
9543:
9538:
9493:
9492:
9483:
9481:
9470:
9423:
9421:
9420:
9415:
9413:
9384:
9383:
9328:
9327:
9288:
9287:
9232:
9231:
9185:
9167:
9165:
9164:
9159:
9157:
9155:
9147:
9130:
9125:
9123:
9115:
9098:
9088:
9068:
9066:
9065:
9060:
9058:
9056:
9048:
9040:
9035:
9033:
9025:
9017:
9006:
9004:
9003:
8998:
8996:
8894:
8892:
8884:
8876:
8836:
8834:
8833:
8828:
8801:
8800:
8791:
8789:
8778:
8733:
8731:
8730:
8725:
8698:
8697:
8688:
8686:
8675:
8629:
8627:
8626:
8621:
8619:
8590:
8589:
8552:
8551:
8536:
8513:
8512:
8475:
8474:
8424:
8422:
8421:
8416:
8414:
8412:
8404:
8396:
8391:
8389:
8381:
8373:
8362:
8360:
8359:
8354:
8352:
8250:
8248:
8240:
8232:
8173:
8171:
8170:
8165:
8163:
8161:
8123:
8102:
8100:
8099:
8049:
8047:
8046:
8041:
8039:
7958:
7956:
7948:
7940:
7882:
7880:
7879:
7874:
7872:
7870:
7850:
7842:
7840:
7839:
7835:
7777:
7775:
7774:
7769:
7767:
7763:
7755:
7743:
7741:
7733:
7725:
7708:Solution method
7702:
7701:
7691:
7689:
7688:
7683:
7656:
7655:
7624:
7623:
7602:
7600:
7599:
7594:
7592:
7508:
7506:
7498:
7490:
7453:
7451:
7450:
7445:
7437:
7435:
7421:
7413:
7411:
7410:
7376:
7374:
7373:
7368:
7366:
7307:
7305:
7297:
7289:
7268:
7266:
7265:
7260:
7233:
7232:
7205:
7203:
7202:
7197:
7195:
7136:
7134:
7126:
7118:
7097:
7095:
7094:
7089:
7074:
7072:
7062:
7061:
7051:
7041:
7040:
7030:
7028:
7027:
7008:
7006:
6996:
6995:
6985:
6975:
6974:
6964:
6962:
6961:
6927:
6925:
6924:
6919:
6917:
6887:
6886:
6868:
6867:
6839:
6838:
6820:
6819:
6796:
6794:
6786:
6778:
6766:
6765:
6747:
6746:
6725:
6724:
6706:
6705:
6669:Solution method
6663:
6662:
6654:
6640:
6637:with respect to
6636:
6625:
6599:
6582:
6576:
6570:
6564:
6554:
6539:
6524:
6513:
6502:
6491:
6469:
6467:
6466:
6461:
6450:
6449:
6431:
6430:
6406:
6405:
6387:
6386:
6368:
6367:
6349:
6348:
6330:
6318:
6317:
6299:
6298:
6283:
6265:
6263:
6262:
6257:
6252:
6238:
6230:
6226:
6207:
6205:
6204:
6199:
6197:
6187:
6186:
6168:
6167:
6135:
6122:
6121:
6101:
6083:
6077:
6073:
6072:
6052:
6039:
6038:
6018:
5980:
5978:
5977:
5972:
5966:
5965:
5941:
5940:
5914:
5912:
5911:
5906:
5904:
5903:
5885:
5881:
5880:
5879:
5856:
5847:
5843:
5832:
5767:
5765:
5764:
5759:
5729:
5728:
5727:
5697:
5695:
5694:
5689:
5687:
5671:
5669:
5668:
5663:
5655:
5654:
5645:
5634:
5633:
5618:
5599:
5597:
5596:
5591:
5589:
5588:
5576:
5575:
5564:
5560:
5550:
5548:
5547:
5535:
5530:
5529:
5505:
5504:
5493:
5489:
5488:
5486:
5485:
5473:
5468:
5467:
5434:
5433:
5422:
5396:
5394:
5393:
5388:
5386:
5384:
5377:
5376:
5358:
5357:
5344:
5343:
5334:
5307:
5305:
5304:
5299:
5297:
5267:This means that
5263:
5261:
5260:
5255:
5253:
5252:
5240:
5200:
5198:
5197:
5192:
5190:
5168:
5166:
5165:
5160:
5158:
5157:
5149:
5137:is defined, and
5129:
5127:
5126:
5121:
5119:
5118:
5110:
5099:
5085:
5084:
5083:
5054:
5052:
5051:
5046:
5019:
5018:
5017:
4986:
4984:
4983:
4978:
4967:
4966:
4947:
4945:
4944:
4939:
4937:
4936:
4913:
4911:
4910:
4905:
4903:
4902:
4890:
4889:
4862:
4854:
4853:
4838:
4837:
4825:
4824:
4809:
4808:
4744:
4734:
4732:
4731:
4726:
4715:
4714:
4696:
4695:
4671:
4670:
4652:
4651:
4625:closed intervals
4618:
4614:
4584:
4582:
4581:
4576:
4565:
4564:
4546:
4545:
4521:
4520:
4502:
4501:
4464:
4462:
4461:
4456:
4451:
4450:
4432:
4431:
4396:
4314:
4313:
4177:Frobenius method
4121:of ordinary and
4103:
4101:
4100:
4095:
4093:
4092:
4087:
4083:
4082:
4078:
4077:
4069:
4053:
4045:
4031:
4023:
3993:
3991:
3990:
3985:
3960:
3958:
3950:
3945:
3934:
3931:
3923:
3841:general solution
3826:maximal solution
3820:
3818:
3817:
3812:
3759:
3737:
3719:
3698:
3696:
3695:
3690:
3666:
3665:
3648:
3639:
3635:
3571:
3550:
3548:
3547:
3542:
3534:
3530:
3529:
3528:
3504:
3434:
3432:
3431:
3426:
3424:
3422:
3421:
3412:
3408:
3400:
3386:
3377:
3365:
3363:
3362:
3357:
3355:
3347:
3343:
3342:
3338:
3329:
3313:
3298:
3296:
3295:
3290:
3288:
3287:
3245:
3244:
3234:
3233:
3222:
3207:
3203:
3194:
3190:
3181:
3167:
3166:
3143:
3142:
3131:
3116:
3112:
3103:
3099:
3090:
3076:
3075:
3059:
3058:
3047:
3032:
3028:
3019:
3015:
3006:
2992:
2991:
2953:
2951:
2950:
2945:
2943:
2935:
2931:
2930:
2929:
2918:
2903:
2899:
2890:
2886:
2877:
2861:
2839:
2837:
2836:
2831:
2829:
2828:
2821:
2817:
2816:
2815:
2798:
2783:
2779:
2770:
2766:
2757:
2741:
2740:
2720:
2716:
2715:
2714:
2697:
2682:
2678:
2669:
2665:
2656:
2640:
2639:
2626:
2622:
2621:
2620:
2603:
2588:
2584:
2575:
2571:
2562:
2546:
2545:
2525:
2524:
2516:
2505:
2484:
2473:
2459:
2448:
2396:
2394:
2393:
2388:
2386:
2382:
2381:
2380:
2363:
2348:
2344:
2335:
2331:
2322:
2306:
2298:
2297:
2286:
2220:
2195:
2187:trivial solution
2184:
2155:) is called the
2146:
2142:
2131:
2125:
2109:
2107:
2106:
2101:
2084:
2083:
2059:
2058:
2048:
2037:
2019:
2018:
1994:
1984:
1982:
1981:
1976:
1950:
1926:
1924:
1923:
1918:
1910:
1906:
1905:
1904:
1887:
1878:
1874:
1863:
1826:takes the form:
1804:
1802:
1801:
1796:
1794:
1793:
1775:
1771:
1770:
1769:
1739:
1690:, a function of
1671:
1669:
1668:
1663:
1658:
1656:
1642:
1637:
1636:
1628:
1622:
1621:
1613:
1593:
1570:
1568:
1566:
1565:
1560:
1557:
1545:
1543:
1542:
1537:
1534:
1522:
1520:
1519:
1514:
1511:
1486:
1408:
1406:
1405:
1400:
1373:
1371:
1370:
1369:
1360:
1354:
1341:
1340:
1335:
1328:
1277:Bernoulli family
1215:natural sciences
1180:Riccati equation
1154:
1151:of the variable
1150:
1146:
1144:
1142:
1141:
1136:
1134:
1133:
1109:
1088:
1086:
1084:
1083:
1078:
1055:
1053:
1051:
1050:
1045:
1034:
1033:
1015:
1013:
1011:
1010:
1005:
994:
993:
972:
970:
969:
964:
938:
937:
913:
912:
894:
877:
876:
864:
847:
846:
822:
821:
731:
724:
717:
695:Phyllis Nicolson
680:Rudolf Lipschitz
563:Green's function
539:Infinite element
530:
495:Solution methods
473:
331:
242:By variable type
196:
195:
78:Natural sciences
71:
70:
50:
49:
42:launched from a
21:
14389:
14388:
14384:
14383:
14382:
14380:
14379:
14378:
14359:
14358:
14357:
14352:
14341:
14290:P-adic analysis
14241:
14227:Matrix calculus
14222:Tensor calculus
14217:Vector calculus
14180:Differentiation
14160:
14154:
14124:
14119:
14060:Jacob Bernoulli
14033:
13998:
13989:Galerkin method
13912:
13850:Solution topics
13845:
13803:
13769:
13708:
13640:
13635:
13529:
13526:
13515:D. Zwillinger,
13492:
13459:
13427:Witold Hurewicz
13413:
13397:Ince, Edward L.
13370:
13342:Hartman, Philip
13316:
13307:
13289:
13265:
13253:
13193:
13171:Kreyszig, Erwin
13164:
13142:
13124:Resnick, Robert
13120:Halliday, David
13115:
13110:
13105:
13098:
13086:
13082:
13077:
13070:
13065:
13061:
13049:
13026:
13016:
13011:
13007:
13002:
12998:
12988:
12983:
12979:
12971:
12967:
12936:
12932:
12924:
12920:
12912:
12908:
12898:
12893:
12889:
12882:
12868:
12864:
12854:
12849:
12845:
12837:
12833:
12825:
12821:
12813:
12809:
12801:
12794:
12786:
12782:
12774:
12770:
12762:
12755:
12747:
12743:
12735:
12731:
12723:
12719:
12711:
12707:
12701:Macmillan Press
12698:
12694:
12677:
12673:
12666:
12652:
12648:
12603:
12599:
12582:
12578:
12569:
12567:
12555:
12554:
12550:
12540:
12538:
12531:
12515:
12511:
12507:
12465:
12368:
12348:
12340:
12332:
12330:
12327:
12326:
12300:
12296:
12285:
12282:
12281:
12273:
12262:
12256:
12253:
12238:
12222:
12211:
12205:
12194:
12167:
12163:
12151:
12147:
12127:
12123:
12122:
12118:
12112:
12108:
12094:
12090:
12089:
12085:
12079:
12075:
12073:
12070:
12069:
12066:Euler's formula
12063:
12054:
12041:
12012:
12008:
12007:
12003:
11986:
11982:
11970:
11966:
11958:
11954:
11953:
11942:
11937:
11933:
11927:
11916:
11903:
11899:
11897:
11894:
11893:
11890:
11884:
11855:
11851:
11850:
11846:
11840:
11836:
11830:
11819:
11806:
11802:
11800:
11797:
11796:
11795:all different,
11794:
11761:
11757:
11742:
11731:
11725:
11722:
11721:
11708:
11683:
11679:
11670:
11666:
11658:
11655:
11654:
11636:
11608:
11604:
11603:
11599:
11597:
11594:
11593:
11580:
11575:
11572:
11530:
11526:
11522:
11512:
11508:
11507:
11505:
11499:
11495:
11489:
11478:
11472:
11469:
11468:
11426:
11422:
11410:
11405:
11401:
11389:
11385:
11365:
11361:
11349:
11344:
11340:
11328:
11324:
11323:
11319:
11304:
11302:
11298:
11294:
11285:
11281:
11279:
11276:
11275:
11263:
11235:
11233:
11229:
11225:
11216:
11212:
11200:
11196:
11184:
11180:
11178:
11175:
11174:
11162:
11125:
11121:
11119:
11112:
11108:
11097:
11093:
11089:
11083:
11079:
11052:
11048:
11046:
11039:
11035:
11024:
11020:
11016:
11010:
11006:
10997:
10993:
10991:
10988:
10987:
10975:
10957:
10953:
10944:
10940:
10932:
10929:
10928:
10910:
10882:
10878:
10877:
10873:
10871:
10868:
10867:
10854:
10849:
10846:
10794:
10786:
10784:
10769:
10765:
10761:
10751:
10747:
10746:
10744:
10742:
10739:
10738:
10711:
10707:
10695:
10691:
10631:
10627:
10626:
10622:
10616:
10612:
10611:
10607:
10601:
10597:
10596:
10592:
10565:
10561:
10557:
10553:
10545:
10542:
10541:
10502:
10498:
10497:
10493:
10491:
10488:
10487:
10433:
10424:
10420:
10410:
10406:
10393:
10385:
10383:
10356:
10352:
10348:
10338:
10334:
10333:
10331:
10329:
10326:
10325:
10251:
10247:
10246:
10242:
10236:
10232:
10231:
10227:
10200:
10196:
10192:
10188:
10180:
10177:
10176:
10132:
10128:
10127:
10123:
10121:
10118:
10117:
10060:
10052:
10050:
10048:
10045:
10044:
10026:
9998:
9994:
9982:
9978:
9950:
9946:
9934:
9932:
9926:
9922:
9911:
9908:
9907:
9886:
9872:
9864:
9862:
9858:
9857:
9847:
9842:
9830:
9826:
9822:
9812:
9808:
9807:
9805:
9795:
9787:
9785:
9780:
9777:
9776:
9775:
9764:
9726:
9722:
9718:
9708:
9704:
9703:
9701:
9699:
9696:
9695:
9677:
9609:
9605:
9595:
9590:
9552:
9549:
9548:
9488:
9484:
9474:
9469:
9431:
9428:
9427:
9411:
9410:
9379:
9375:
9323:
9319:
9315:
9309:
9308:
9283:
9279:
9227:
9223:
9219:
9194:
9192:
9189:
9188:
9172:
9148:
9131:
9129:
9116:
9099:
9097:
9095:
9092:
9091:
9075:
9049:
9041:
9039:
9026:
9018:
9016:
9014:
9011:
9010:
8994:
8993:
8983:
8927:
8926:
8916:
8885:
8877:
8875:
8853:
8851:
8848:
8847:
8796:
8792:
8782:
8777:
8739:
8736:
8735:
8693:
8689:
8679:
8674:
8636:
8633:
8632:
8617:
8616:
8585:
8581:
8547:
8543:
8534:
8533:
8508:
8504:
8470:
8466:
8459:
8437:
8435:
8432:
8431:
8405:
8397:
8395:
8382:
8374:
8372:
8370:
8367:
8366:
8350:
8349:
8339:
8283:
8282:
8272:
8241:
8233:
8231:
8209:
8207:
8204:
8203:
8124:
8103:
8101:
8092:
8088:
8065:
8062:
8061:
8037:
8036:
8026:
7970:
7969:
7959:
7949:
7941:
7939:
7895:
7893:
7890:
7889:
7851:
7843:
7841:
7831:
7827:
7823:
7800:
7797:
7796:
7754:
7750:
7734:
7726:
7724:
7722:
7719:
7718:
7700:
7651:
7647:
7619:
7615:
7613:
7610:
7609:
7590:
7589:
7579:
7535:
7534:
7524:
7499:
7491:
7489:
7473:
7471:
7468:
7467:
7422:
7414:
7412:
7406:
7402:
7394:
7391:
7390:
7364:
7363:
7337:
7328:
7327:
7308:
7298:
7290:
7288:
7284:
7282:
7279:
7278:
7228:
7224:
7216:
7213:
7212:
7193:
7192:
7166:
7157:
7156:
7137:
7127:
7119:
7117:
7113:
7111:
7108:
7107:
7057:
7053:
7052:
7036:
7032:
7031:
7029:
7023:
7019:
6991:
6987:
6986:
6970:
6966:
6965:
6963:
6957:
6953:
6951:
6948:
6947:
6943:
6937:
6915:
6914:
6904:
6882:
6878:
6863:
6859:
6834:
6830:
6815:
6811:
6808:
6807:
6797:
6787:
6779:
6777:
6761:
6757:
6742:
6738:
6720:
6716:
6701:
6697:
6693:
6691:
6688:
6687:
6661:
6646:
6638:
6627:
6612:
6597:
6590:
6584:
6578:
6572:
6566:
6560:
6541:
6526:
6515:
6504:
6493:
6482:
6476:
6445:
6441:
6426:
6422:
6401:
6397:
6382:
6378:
6363:
6359:
6344:
6340:
6326:
6313:
6309:
6294:
6290:
6279:
6277:
6274:
6273:
6248:
6234:
6222:
6221:
6219:
6216:
6215:
6195:
6194:
6182:
6178:
6163:
6159:
6145:
6140:
6131:
6124:
6123:
6117:
6113:
6111:
6106:
6091:
6084:
6082:
6075:
6074:
6068:
6064:
6062:
6057:
6048:
6041:
6040:
6034:
6030:
6028:
6023:
6014:
6006:
6004:
6001:
6000:
5949:
5945:
5936:
5932:
5930:
5927:
5926:
5893:
5889:
5863:
5859:
5836:
5825:
5812:
5808:
5803:
5800:
5799:
5789:
5777:
5723:
5719:
5712:
5706:
5703:
5702:
5683:
5681:
5678:
5677:
5650:
5646:
5641:
5629:
5625:
5614:
5612:
5609:
5608:
5584:
5583:
5571:
5567:
5565:
5543:
5539:
5534:
5525:
5521:
5520:
5516:
5513:
5512:
5500:
5496:
5494:
5481:
5477:
5472:
5463:
5459:
5449:
5445:
5442:
5441:
5429:
5425:
5423:
5418:
5411:
5410:
5408:
5405:
5404:
5372:
5368:
5353:
5349:
5345:
5339:
5335:
5333:
5316:
5313:
5312:
5293:
5291:
5288:
5287:
5248:
5244:
5233:
5231:
5228:
5227:
5216:
5209:
5186:
5184:
5181:
5180:
5148:
5147:
5142:
5139:
5138:
5109:
5108:
5079:
5075:
5068:
5062:
5059:
5058:
5013:
5009:
5002:
4996:
4993:
4992:
4962:
4958:
4956:
4953:
4952:
4932:
4928:
4926:
4923:
4922:
4898:
4894:
4885:
4881:
4858:
4849:
4845:
4833:
4829:
4820:
4816:
4804:
4800:
4798:
4795:
4794:
4789:
4782:
4771:
4736:
4710:
4706:
4691:
4687:
4666:
4662:
4647:
4643:
4632:
4629:
4628:
4616:
4602:
4601:(symbolically:
4560:
4556:
4541:
4537:
4516:
4512:
4497:
4493:
4482:
4479:
4478:
4446:
4442:
4427:
4423:
4389:
4387:
4384:
4383:
4380:
4303:
4275:
4269:
4207:
4181:Two memoirs by
4179:
4173:
4171:Fuchsian theory
4160:complex numbers
4144:
4115:
4110:
4088:
4068:
4061:
4057:
4044:
4037:
4033:
4032:
4022:
4005:
4002:
4001:
3954:
3946:
3944:
3930:
3916:
3914:
3911:
3910:
3900:
3884:guessing method
3834:global solution
3768:
3765:
3764:
3751:
3721:
3703:
3655:
3651:
3628:
3608:
3605:
3604:
3555:
3518:
3514:
3497:
3484:
3480:
3475:
3472:
3471:
3465:
3417:
3413:
3404:
3396:
3382:
3378:
3376:
3374:
3371:
3370:
3368:Jacobian matrix
3351:
3334:
3333:
3325:
3318:
3314:
3309:
3307:
3304:
3303:
3282:
3281:
3275:
3274:
3268:
3267:
3261:
3260:
3250:
3249:
3239:
3238:
3223:
3218:
3217:
3199:
3198:
3186:
3185:
3177:
3162:
3158:
3155:
3154:
3148:
3147:
3132:
3127:
3126:
3108:
3107:
3095:
3094:
3086:
3071:
3067:
3064:
3063:
3048:
3043:
3042:
3024:
3023:
3011:
3010:
3002:
2987:
2983:
2976:
2975:
2973:
2970:
2969:
2939:
2919:
2914:
2913:
2895:
2894:
2882:
2881:
2873:
2866:
2862:
2857:
2855:
2852:
2851:
2823:
2822:
2799:
2794:
2793:
2775:
2774:
2762:
2761:
2753:
2746:
2742:
2736:
2732:
2729:
2728:
2722:
2721:
2698:
2693:
2692:
2674:
2673:
2661:
2660:
2652:
2645:
2641:
2635:
2631:
2628:
2627:
2604:
2599:
2598:
2580:
2579:
2567:
2566:
2558:
2551:
2547:
2541:
2537:
2530:
2529:
2519:
2518:
2506:
2501:
2494:
2493:
2487:
2486:
2474:
2469:
2462:
2461:
2449:
2444:
2433:
2432:
2430:
2427:
2426:
2364:
2359:
2358:
2340:
2339:
2327:
2326:
2318:
2311:
2307:
2302:
2287:
2282:
2281:
2279:
2276:
2275:
2245:
2239:
2229:
2211:
2203:
2190:
2175:
2167:
2147:. The function
2144:
2133:
2123:
2122:
2113:
2073:
2069:
2054:
2050:
2038:
2027:
2008:
2004:
2002:
1999:
1998:
1990:
1970:
1967:
1966:
1958:
1946:
1938:
1894:
1890:
1867:
1856:
1843:
1839:
1834:
1831:
1830:
1783:
1779:
1753:
1749:
1732:
1719:
1715:
1710:
1707:
1706:
1684:
1678:
1646:
1641:
1627:
1626:
1612:
1611:
1606:
1603:
1602:
1598:compactly, and
1579:
1561:
1558:
1550:
1549:
1547:
1538:
1535:
1527:
1526:
1524:
1515:
1512:
1507:
1506:
1504:
1502:
1473:
1462:
1365:
1361:
1356:
1355:
1336:
1331:
1330:
1329:
1327:
1322:
1319:
1318:
1203:
1152:
1148:
1123:
1119:
1102:
1100:
1097:
1096:
1094:
1063:
1060:
1059:
1057:
1029:
1025:
1023:
1020:
1019:
1017:
989:
985:
983:
980:
979:
977:
927:
923:
908:
904:
887:
872:
868:
857:
842:
838:
817:
813:
811:
808:
807:
789:
735:
706:
705:
704:
635:Jacob Bernoulli
619:
606:
605:
587:
556:Petrov–Galerkin
524:
509:
496:
488:
487:
486:
468:
414:Boundary values
403:
395:
394:
370:
357:
356:
355:
329:
323:
315:
314:
302:
279:
237:
193:
180:
179:
175:
153:Social sciences
109:
87:
68:
28:
23:
22:
15:
12:
11:
5:
14387:
14377:
14376:
14371:
14354:
14353:
14346:
14343:
14342:
14340:
14339:
14334:
14329:
14324:
14319:
14314:
14308:
14307:
14302:
14300:Measure theory
14297:
14294:P-adic numbers
14287:
14282:
14277:
14272:
14267:
14257:
14252:
14246:
14243:
14242:
14240:
14239:
14234:
14229:
14224:
14219:
14214:
14209:
14204:
14203:
14202:
14197:
14192:
14182:
14177:
14165:
14162:
14161:
14153:
14152:
14145:
14138:
14130:
14121:
14120:
14118:
14117:
14112:
14107:
14102:
14097:
14092:
14087:
14082:
14077:
14075:Ernst Lindelöf
14072:
14067:
14062:
14057:
14055:Leonhard Euler
14052:
14047:
14041:
14039:
14038:Mathematicians
14035:
14034:
14032:
14031:
14026:
14021:
14016:
14010:
14008:
14004:
14003:
14000:
13999:
13997:
13996:
13991:
13986:
13981:
13976:
13971:
13966:
13961:
13956:
13951:
13946:
13941:
13936:
13931:
13926:
13920:
13918:
13914:
13913:
13911:
13910:
13905:
13900:
13894:
13889:
13884:
13879:
13874:
13869:
13864:
13862:Phase portrait
13859:
13853:
13851:
13847:
13846:
13844:
13843:
13838:
13833:
13828:
13822:
13820:
13813:
13809:
13808:
13805:
13804:
13802:
13801:
13796:
13795:
13794:
13784:
13777:
13775:
13771:
13770:
13768:
13767:
13765:On jet bundles
13762:
13757:
13752:
13747:
13742:
13737:
13732:
13730:Nonhomogeneous
13727:
13722:
13716:
13714:
13710:
13709:
13707:
13706:
13701:
13696:
13691:
13686:
13681:
13676:
13671:
13666:
13661:
13655:
13653:
13646:
13645:Classification
13642:
13641:
13634:
13633:
13626:
13619:
13611:
13605:
13604:
13599:
13593:
13583:
13573:
13567:
13561:
13551:
13545:
13525:
13524:External links
13522:
13521:
13520:
13513:
13499:A. D. Polyanin
13496:
13490:
13468:Teschl, Gerald
13464:
13457:
13441:
13424:
13411:
13393:
13381:
13368:
13338:
13315:
13312:
13311:
13310:
13305:
13292:
13288:978-0750305303
13287:
13270:
13256:
13251:
13234:
13211:
13197:
13191:
13167:
13162:
13152:, New Jersey:
13145:
13140:
13114:
13111:
13109:
13108:
13096:
13080:
13068:
13059:
13024:
13005:
12996:
12985:Lawrence (1999
12977:
12965:
12930:
12926:Kreyszig (1972
12918:
12914:Kreyszig (1972
12906:
12887:
12880:
12862:
12843:
12839:Kreyszig (1972
12831:
12829:, p. 128)
12819:
12807:
12803:Kreyszig (1972
12792:
12780:
12776:Kreyszig (1972
12768:
12766:, p. 127)
12753:
12741:
12729:
12717:
12713:Kreyszig (1972
12705:
12692:
12671:
12664:
12646:
12597:
12576:
12565:Stack Exchange
12548:
12529:
12508:
12506:
12503:
12502:
12501:
12496:
12491:
12486:
12481:
12476:
12471:
12464:
12461:
12460:
12459:
12453:
12443:
12437:
12431:
12425:
12419:
12413:
12407:
12401:
12395:
12389:
12379:
12367:
12364:
12347:
12339:
12306:
12303:
12299:
12295:
12292:
12289:
12275:
12274:
12225:
12223:
12216:
12207:Main article:
12204:
12201:
12198:
12197:
12190:
12175:
12170:
12166:
12162:
12159:
12154:
12150:
12146:
12143:
12140:
12135:
12130:
12126:
12121:
12115:
12111:
12107:
12102:
12097:
12093:
12088:
12082:
12078:
12059:
12050:
12037:
12020:
12015:
12011:
12006:
12001:
11995:
11992:
11989:
11985:
11979:
11976:
11973:
11969:
11961:
11957:
11951:
11948:
11945:
11941:
11936:
11930:
11925:
11922:
11919:
11915:
11911:
11906:
11902:
11888:
11880:
11876:for each root
11863:
11858:
11854:
11849:
11843:
11839:
11833:
11828:
11825:
11822:
11818:
11814:
11809:
11805:
11790:
11775:
11772:
11769:
11764:
11760:
11756:
11753:
11750:
11745:
11740:
11737:
11734:
11730:
11704:
11686:
11682:
11678:
11673:
11669:
11665:
11662:
11652:
11634:
11616:
11611:
11607:
11602:
11578:
11570:
11565:
11553:
11550:
11547:
11544:
11541:
11533:
11529:
11525:
11520:
11515:
11511:
11502:
11498:
11492:
11487:
11484:
11481:
11477:
11460:
11459:
11446:
11441:
11435:
11429:
11425:
11421:
11418:
11415:
11408:
11404:
11400:
11397:
11392:
11388:
11384:
11380:
11374:
11368:
11364:
11360:
11357:
11354:
11347:
11343:
11339:
11336:
11331:
11327:
11322:
11314:
11310:
11307:
11301:
11297:
11293:
11288:
11284:
11245:
11241:
11238:
11232:
11228:
11224:
11219:
11215:
11211:
11208:
11203:
11199:
11195:
11192:
11187:
11183:
11145:
11139:
11136:
11133:
11128:
11124:
11118:
11115:
11111:
11104:
11101:
11096:
11092:
11086:
11082:
11078:
11072:
11066:
11063:
11060:
11055:
11051:
11045:
11042:
11038:
11031:
11028:
11023:
11019:
11013:
11009:
11005:
11000:
10996:
10960:
10956:
10952:
10947:
10943:
10939:
10936:
10926:
10908:
10890:
10885:
10881:
10876:
10852:
10844:
10839:
10827:
10824:
10821:
10818:
10815:
10812:
10809:
10806:
10800:
10797:
10792:
10789:
10783:
10780:
10772:
10768:
10764:
10759:
10754:
10750:
10733:
10732:
10720:
10714:
10710:
10706:
10703:
10698:
10694:
10690:
10687:
10684:
10680:
10676:
10673:
10669:
10666:
10663:
10660:
10655:
10652:
10648:
10645:
10642:
10639:
10634:
10630:
10625:
10619:
10615:
10610:
10604:
10600:
10595:
10589:
10586:
10582:
10579:
10576:
10573:
10568:
10564:
10560:
10556:
10552:
10549:
10539:
10526:
10523:
10519:
10516:
10513:
10510:
10505:
10501:
10496:
10484:
10472:
10469:
10466:
10463:
10460:
10457:
10453:
10449:
10446:
10443:
10439:
10436:
10432:
10427:
10423:
10419:
10416:
10413:
10409:
10405:
10399:
10396:
10391:
10388:
10382:
10379:
10376:
10373:
10370:
10367:
10359:
10355:
10351:
10346:
10341:
10337:
10320:
10319:
10306:
10302:
10299:
10296:
10293:
10289:
10286:
10283:
10280:
10275:
10272:
10268:
10265:
10262:
10259:
10254:
10250:
10245:
10239:
10235:
10230:
10224:
10221:
10217:
10214:
10211:
10208:
10203:
10199:
10195:
10191:
10187:
10184:
10172:
10161:
10156:
10153:
10149:
10146:
10143:
10140:
10135:
10131:
10126:
10114:
10102:
10099:
10096:
10093:
10090:
10087:
10084:
10081:
10078:
10075:
10072:
10066:
10063:
10058:
10055:
10039:
10038:
10035:
10032:
10025:
10020:Linear to the
10018:
10015:
10014:
10001:
9997:
9993:
9985:
9981:
9977:
9974:
9971:
9967:
9964:
9961:
9958:
9953:
9949:
9945:
9940:
9937:
9929:
9925:
9921:
9918:
9915:
9905:
9889:
9884:
9878:
9875:
9870:
9867:
9861:
9853:
9850:
9846:
9841:
9833:
9829:
9825:
9820:
9815:
9811:
9801:
9798:
9793:
9790:
9784:
9761:
9749:
9746:
9743:
9740:
9737:
9729:
9725:
9721:
9716:
9711:
9707:
9690:
9689:
9686:
9683:
9676:
9673:
9670:
9669:
9657:
9654:
9650:
9647:
9644:
9641:
9638:
9635:
9632:
9629:
9626:
9623:
9620:
9617:
9612:
9608:
9601:
9598:
9594:
9589:
9586:
9583:
9580:
9577:
9574:
9571:
9568:
9565:
9562:
9559:
9556:
9536:
9533:
9529:
9526:
9523:
9520:
9517:
9514:
9511:
9508:
9505:
9502:
9499:
9496:
9491:
9487:
9480:
9477:
9473:
9468:
9465:
9462:
9459:
9456:
9453:
9450:
9447:
9444:
9441:
9438:
9435:
9409:
9406:
9403:
9400:
9396:
9393:
9390:
9387:
9382:
9378:
9374:
9371:
9368:
9364:
9361:
9358:
9355:
9352:
9349:
9346:
9343:
9340:
9337:
9334:
9331:
9326:
9322:
9318:
9316:
9314:
9311:
9310:
9307:
9304:
9300:
9297:
9294:
9291:
9286:
9282:
9278:
9275:
9272:
9268:
9265:
9262:
9259:
9256:
9253:
9250:
9247:
9244:
9241:
9238:
9235:
9230:
9226:
9222:
9220:
9218:
9215:
9212:
9209:
9206:
9203:
9200:
9197:
9196:
9169:
9154:
9151:
9146:
9143:
9140:
9137:
9134:
9128:
9122:
9119:
9114:
9111:
9108:
9105:
9102:
9070:
9055:
9052:
9047:
9044:
9038:
9032:
9029:
9024:
9021:
8992:
8989:
8986:
8984:
8982:
8979:
8975:
8972:
8969:
8966:
8963:
8960:
8957:
8954:
8951:
8947:
8944:
8941:
8938:
8935:
8932:
8929:
8928:
8925:
8922:
8919:
8917:
8915:
8912:
8909:
8906:
8903:
8900:
8897:
8891:
8888:
8883:
8880:
8874:
8871:
8868:
8865:
8862:
8859:
8856:
8855:
8845:, first-order
8839:
8838:
8826:
8823:
8819:
8816:
8813:
8810:
8807:
8804:
8799:
8795:
8788:
8785:
8781:
8776:
8773:
8770:
8767:
8764:
8761:
8758:
8755:
8752:
8749:
8746:
8743:
8723:
8720:
8716:
8713:
8710:
8707:
8704:
8701:
8696:
8692:
8685:
8682:
8678:
8673:
8670:
8667:
8664:
8661:
8658:
8655:
8652:
8649:
8646:
8643:
8640:
8615:
8612:
8609:
8606:
8602:
8599:
8596:
8593:
8588:
8584:
8580:
8577:
8574:
8570:
8567:
8564:
8561:
8558:
8555:
8550:
8546:
8542:
8539:
8537:
8535:
8532:
8529:
8525:
8522:
8519:
8516:
8511:
8507:
8503:
8500:
8497:
8493:
8490:
8487:
8484:
8481:
8478:
8473:
8469:
8465:
8462:
8460:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8439:
8429:
8426:
8411:
8408:
8403:
8400:
8394:
8388:
8385:
8380:
8377:
8348:
8345:
8342:
8340:
8338:
8335:
8331:
8328:
8325:
8322:
8319:
8316:
8313:
8310:
8307:
8303:
8300:
8297:
8294:
8291:
8288:
8285:
8284:
8281:
8278:
8275:
8273:
8271:
8268:
8265:
8262:
8259:
8256:
8253:
8247:
8244:
8239:
8236:
8230:
8227:
8224:
8221:
8218:
8215:
8212:
8211:
8201:, first-order
8195:
8194:
8160:
8157:
8154:
8151:
8148:
8145:
8142:
8139:
8136:
8133:
8130:
8127:
8122:
8119:
8115:
8112:
8109:
8106:
8098:
8095:
8091:
8087:
8084:
8081:
8078:
8075:
8072:
8069:
8058:
8051:
8035:
8032:
8029:
8027:
8025:
8022:
8018:
8015:
8012:
8009:
8006:
8003:
8000:
7997:
7994:
7990:
7987:
7984:
7981:
7978:
7975:
7972:
7971:
7968:
7965:
7962:
7960:
7955:
7952:
7947:
7944:
7937:
7934:
7931:
7928:
7925:
7922:
7919:
7916:
7913:
7910:
7907:
7904:
7901:
7898:
7897:
7884:
7883:
7869:
7866:
7863:
7860:
7857:
7854:
7849:
7846:
7838:
7834:
7830:
7826:
7822:
7819:
7816:
7813:
7810:
7807:
7804:
7794:
7779:
7766:
7761:
7758:
7753:
7749:
7746:
7740:
7737:
7732:
7729:
7713:
7712:
7709:
7706:
7699:
7696:
7693:
7692:
7681:
7678:
7675:
7672:
7668:
7665:
7662:
7659:
7654:
7650:
7646:
7643:
7640:
7636:
7633:
7630:
7627:
7622:
7618:
7607:
7604:
7588:
7585:
7582:
7580:
7578:
7575:
7571:
7568:
7565:
7562:
7559:
7556:
7553:
7549:
7546:
7543:
7540:
7537:
7536:
7533:
7530:
7527:
7525:
7523:
7520:
7517:
7514:
7511:
7505:
7502:
7497:
7494:
7488:
7485:
7482:
7479:
7476:
7475:
7455:
7454:
7443:
7440:
7434:
7431:
7428:
7425:
7420:
7417:
7409:
7405:
7401:
7398:
7388:
7378:
7362:
7359:
7355:
7352:
7349:
7346:
7343:
7340:
7338:
7336:
7333:
7330:
7329:
7326:
7323:
7320:
7317:
7314:
7311:
7309:
7304:
7301:
7296:
7293:
7287:
7286:
7270:
7269:
7258:
7255:
7252:
7249:
7245:
7242:
7239:
7236:
7231:
7227:
7223:
7220:
7210:
7207:
7191:
7188:
7184:
7181:
7178:
7175:
7172:
7169:
7167:
7165:
7162:
7159:
7158:
7155:
7152:
7149:
7146:
7143:
7140:
7138:
7133:
7130:
7125:
7122:
7116:
7115:
7099:
7098:
7087:
7084:
7081:
7078:
7071:
7068:
7065:
7060:
7056:
7050:
7047:
7044:
7039:
7035:
7026:
7022:
7018:
7015:
7012:
7005:
7002:
6999:
6994:
6990:
6984:
6981:
6978:
6973:
6969:
6960:
6956:
6945:
6941:
6935:
6929:
6913:
6910:
6907:
6905:
6903:
6900:
6896:
6893:
6890:
6885:
6881:
6877:
6874:
6871:
6866:
6862:
6858:
6855:
6852:
6848:
6845:
6842:
6837:
6833:
6829:
6826:
6823:
6818:
6814:
6810:
6809:
6806:
6803:
6800:
6798:
6793:
6790:
6785:
6782:
6775:
6772:
6769:
6764:
6760:
6756:
6753:
6750:
6745:
6741:
6737:
6734:
6731:
6728:
6723:
6719:
6715:
6712:
6709:
6704:
6700:
6696:
6695:
6674:
6673:
6670:
6667:
6660:
6657:
6595:
6588:
6475:
6472:
6471:
6470:
6459:
6456:
6453:
6448:
6444:
6440:
6437:
6434:
6429:
6425:
6421:
6418:
6415:
6412:
6409:
6404:
6400:
6396:
6393:
6390:
6385:
6381:
6377:
6374:
6371:
6366:
6362:
6358:
6355:
6352:
6347:
6343:
6339:
6336:
6333:
6329:
6324:
6321:
6316:
6312:
6308:
6305:
6302:
6297:
6293:
6289:
6286:
6282:
6267:
6266:
6255:
6251:
6247:
6244:
6241:
6237:
6233:
6229:
6225:
6209:
6208:
6193:
6190:
6185:
6181:
6177:
6174:
6171:
6166:
6162:
6158:
6155:
6152:
6149:
6146:
6144:
6141:
6138:
6134:
6130:
6126:
6125:
6120:
6116:
6112:
6110:
6107:
6104:
6100:
6097:
6094:
6090:
6086:
6085:
6081:
6078:
6076:
6071:
6067:
6063:
6061:
6058:
6055:
6051:
6047:
6043:
6042:
6037:
6033:
6029:
6027:
6024:
6021:
6017:
6013:
6009:
6008:
5982:
5981:
5969:
5964:
5961:
5958:
5955:
5952:
5948:
5944:
5939:
5935:
5916:
5915:
5902:
5899:
5896:
5892:
5888:
5884:
5878:
5875:
5872:
5869:
5866:
5862:
5855:
5852:
5846:
5842:
5839:
5835:
5831:
5828:
5824:
5821:
5818:
5815:
5811:
5807:
5788:
5785:
5776:
5773:
5769:
5768:
5757:
5754:
5751:
5748:
5745:
5742:
5739:
5736:
5733:
5726:
5722:
5718:
5715:
5711:
5686:
5661:
5658:
5653:
5649:
5644:
5640:
5637:
5632:
5628:
5624:
5621:
5617:
5601:
5600:
5587:
5582:
5579:
5574:
5570:
5566:
5563:
5559:
5556:
5553:
5546:
5542:
5538:
5533:
5528:
5524:
5519:
5515:
5514:
5511:
5508:
5503:
5499:
5495:
5492:
5484:
5480:
5476:
5471:
5466:
5462:
5458:
5455:
5452:
5448:
5444:
5443:
5440:
5437:
5432:
5428:
5424:
5421:
5417:
5416:
5414:
5398:
5397:
5383:
5380:
5375:
5371:
5367:
5364:
5361:
5356:
5352:
5348:
5342:
5338:
5332:
5329:
5326:
5323:
5320:
5296:
5265:
5264:
5251:
5247:
5243:
5239:
5236:
5224:
5223:
5219:
5218:
5214:
5207:
5201:
5189:
5177:
5155:
5152:
5146:
5131:
5130:
5116:
5113:
5107:
5104:
5098:
5095:
5092:
5089:
5082:
5078:
5074:
5071:
5067:
5055:
5044:
5041:
5038:
5035:
5032:
5029:
5026:
5023:
5016:
5012:
5008:
5005:
5001:
5000:lim sup
4976:
4973:
4970:
4965:
4961:
4935:
4931:
4915:
4914:
4901:
4897:
4893:
4888:
4884:
4880:
4877:
4874:
4871:
4868:
4865:
4861:
4857:
4852:
4848:
4844:
4841:
4836:
4832:
4828:
4823:
4819:
4815:
4812:
4807:
4803:
4787:
4780:
4770:
4767:
4724:
4721:
4718:
4713:
4709:
4705:
4702:
4699:
4694:
4690:
4686:
4683:
4680:
4677:
4674:
4669:
4665:
4661:
4658:
4655:
4650:
4646:
4642:
4639:
4636:
4574:
4571:
4568:
4563:
4559:
4555:
4552:
4549:
4544:
4540:
4536:
4533:
4530:
4527:
4524:
4519:
4515:
4511:
4508:
4505:
4500:
4496:
4492:
4489:
4486:
4454:
4449:
4445:
4441:
4438:
4435:
4430:
4426:
4421:
4417:
4414:
4411:
4408:
4405:
4402:
4399:
4395:
4392:
4379:
4376:
4361:
4360:
4357:
4356:
4353:
4347:
4341:
4340:
4337:
4331:
4325:
4324:
4321:
4318:
4302:
4299:
4283:eigenfunctions
4271:Main article:
4268:
4265:
4240:). Continuous
4206:
4203:
4175:Main article:
4172:
4169:
4143:
4140:
4117:The theory of
4114:
4111:
4109:
4106:
4105:
4104:
4091:
4086:
4081:
4075:
4072:
4067:
4064:
4060:
4056:
4051:
4048:
4043:
4040:
4036:
4029:
4026:
4021:
4018:
4015:
4012:
4009:
3995:
3994:
3983:
3980:
3977:
3974:
3971:
3968:
3963:
3957:
3953:
3949:
3943:
3940:
3937:
3929:
3926:
3922:
3919:
3899:
3896:
3822:
3821:
3809:
3806:
3803:
3800:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3700:
3699:
3688:
3685:
3682:
3679:
3675:
3672:
3669:
3664:
3661:
3658:
3654:
3647:
3644:
3638:
3634:
3631:
3627:
3624:
3621:
3618:
3615:
3612:
3582:integral curve
3552:
3551:
3540:
3537:
3533:
3527:
3524:
3521:
3517:
3513:
3510:
3507:
3503:
3500:
3496:
3493:
3490:
3487:
3483:
3479:
3464:
3461:
3457:phase portrait
3446:Hessian matrix
3420:
3416:
3411:
3407:
3403:
3399:
3395:
3392:
3389:
3385:
3381:
3354:
3350:
3346:
3341:
3337:
3332:
3328:
3324:
3321:
3317:
3312:
3300:
3299:
3286:
3280:
3277:
3276:
3273:
3270:
3269:
3266:
3263:
3262:
3259:
3256:
3255:
3253:
3248:
3243:
3237:
3232:
3229:
3226:
3221:
3216:
3213:
3210:
3206:
3202:
3197:
3193:
3189:
3184:
3180:
3176:
3173:
3170:
3165:
3161:
3157:
3156:
3153:
3150:
3149:
3146:
3141:
3138:
3135:
3130:
3125:
3122:
3119:
3115:
3111:
3106:
3102:
3098:
3093:
3089:
3085:
3082:
3079:
3074:
3070:
3066:
3065:
3062:
3057:
3054:
3051:
3046:
3041:
3038:
3035:
3031:
3027:
3022:
3018:
3014:
3009:
3005:
3001:
2998:
2995:
2990:
2986:
2982:
2981:
2979:
2955:
2954:
2942:
2938:
2934:
2928:
2925:
2922:
2917:
2912:
2909:
2906:
2902:
2898:
2893:
2889:
2885:
2880:
2876:
2872:
2869:
2865:
2860:
2841:
2840:
2827:
2820:
2814:
2811:
2808:
2805:
2802:
2797:
2792:
2789:
2786:
2782:
2778:
2773:
2769:
2765:
2760:
2756:
2752:
2749:
2745:
2739:
2735:
2731:
2730:
2727:
2724:
2723:
2719:
2713:
2710:
2707:
2704:
2701:
2696:
2691:
2688:
2685:
2681:
2677:
2672:
2668:
2664:
2659:
2655:
2651:
2648:
2644:
2638:
2634:
2630:
2629:
2625:
2619:
2616:
2613:
2610:
2607:
2602:
2597:
2594:
2591:
2587:
2583:
2578:
2574:
2570:
2565:
2561:
2557:
2554:
2550:
2544:
2540:
2536:
2535:
2533:
2528:
2523:
2515:
2512:
2509:
2504:
2500:
2496:
2495:
2492:
2489:
2488:
2483:
2480:
2477:
2472:
2468:
2464:
2463:
2458:
2455:
2452:
2447:
2443:
2439:
2438:
2436:
2398:
2397:
2385:
2379:
2376:
2373:
2370:
2367:
2362:
2357:
2354:
2351:
2347:
2343:
2338:
2334:
2330:
2325:
2321:
2317:
2314:
2310:
2305:
2301:
2296:
2293:
2290:
2285:
2241:Main article:
2238:
2237:System of ODEs
2235:
2234:
2233:
2230:
2225:
2223:
2222:
2208:nonhomogeneous
2204:
2199:
2197:
2168:
2163:
2160:
2118:
2111:
2110:
2099:
2096:
2093:
2090:
2087:
2082:
2079:
2076:
2072:
2068:
2065:
2062:
2057:
2053:
2047:
2044:
2041:
2036:
2033:
2030:
2026:
2022:
2017:
2014:
2011:
2007:
1974:
1959:
1954:
1952:
1939:
1934:
1928:
1927:
1916:
1913:
1909:
1903:
1900:
1897:
1893:
1886:
1883:
1877:
1873:
1870:
1866:
1862:
1859:
1855:
1852:
1849:
1846:
1842:
1838:
1806:
1805:
1792:
1789:
1786:
1782:
1778:
1774:
1768:
1765:
1762:
1759:
1756:
1752:
1748:
1745:
1742:
1738:
1735:
1731:
1728:
1725:
1722:
1718:
1714:
1677:
1674:
1661:
1655:
1652:
1649:
1645:
1640:
1634:
1631:
1625:
1619:
1616:
1610:
1461:
1458:
1420:. In general,
1410:
1409:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1368:
1364:
1359:
1353:
1350:
1347:
1344:
1339:
1334:
1326:
1202:
1199:
1132:
1129:
1126:
1122:
1118:
1115:
1112:
1108:
1105:
1089:are arbitrary
1076:
1073:
1070:
1067:
1043:
1040:
1037:
1032:
1028:
1003:
1000:
997:
992:
988:
974:
973:
962:
959:
956:
953:
950:
947:
944:
941:
936:
933:
930:
926:
922:
919:
916:
911:
907:
903:
900:
897:
893:
890:
886:
883:
880:
875:
871:
867:
863:
860:
856:
853:
850:
845:
841:
837:
834:
831:
828:
825:
820:
816:
788:
785:
775:
737:
736:
734:
733:
726:
719:
711:
708:
707:
703:
702:
697:
692:
687:
685:Ernst Lindelöf
682:
677:
672:
667:
662:
657:
655:Joseph Fourier
652:
647:
642:
640:Leonhard Euler
637:
632:
627:
621:
620:
617:
616:
613:
612:
608:
607:
604:
603:
598:
593:
586:
585:
580:
575:
570:
565:
560:
559:
558:
548:
543:
542:
541:
534:Finite element
531:
527:Crank–Nicolson
518:
513:
507:
502:
498:
497:
494:
493:
490:
489:
485:
484:
479:
474:
466:
461:
448:
446:Phase portrait
443:
438:
437:
436:
434:Cauchy problem
431:
426:
421:
411:
405:
404:
402:General topics
401:
400:
397:
396:
393:
392:
387:
382:
377:
371:
368:
367:
364:
363:
359:
358:
354:
353:
348:
347:
346:
335:
334:
333:
324:
321:
320:
317:
316:
311:
310:
309:
308:
301:
300:
295:
289:
286:
285:
281:
280:
278:
277:
275:Nonhomogeneous
268:
263:
260:
254:
253:
252:
244:
243:
239:
238:
236:
235:
230:
225:
220:
215:
210:
205:
199:
194:
191:
190:
187:
186:
185:Classification
182:
181:
172:
171:
170:
169:
164:
156:
155:
149:
148:
147:
146:
141:
136:
128:
127:
121:
120:
119:
118:
113:
107:
102:
97:
89:
88:
86:
85:
80:
74:
69:
66:
65:
62:
61:
57:
56:
26:
9:
6:
4:
3:
2:
14386:
14375:
14372:
14370:
14367:
14366:
14364:
14351:
14350:
14344:
14338:
14335:
14333:
14330:
14328:
14325:
14323:
14320:
14318:
14315:
14313:
14310:
14309:
14306:
14303:
14301:
14298:
14295:
14291:
14288:
14286:
14283:
14281:
14278:
14276:
14273:
14271:
14268:
14265:
14261:
14258:
14256:
14253:
14251:
14250:Real analysis
14248:
14247:
14244:
14238:
14235:
14233:
14230:
14228:
14225:
14223:
14220:
14218:
14215:
14213:
14210:
14208:
14205:
14201:
14198:
14196:
14193:
14191:
14188:
14187:
14186:
14183:
14181:
14178:
14176:
14172:
14171:
14167:
14166:
14163:
14159:
14151:
14146:
14144:
14139:
14137:
14132:
14131:
14128:
14116:
14113:
14111:
14108:
14106:
14103:
14101:
14098:
14096:
14093:
14091:
14088:
14086:
14083:
14081:
14078:
14076:
14073:
14071:
14068:
14066:
14063:
14061:
14058:
14056:
14053:
14051:
14048:
14046:
14043:
14042:
14040:
14036:
14030:
14027:
14025:
14022:
14020:
14017:
14015:
14012:
14011:
14009:
14005:
13995:
13992:
13990:
13987:
13985:
13982:
13980:
13977:
13975:
13972:
13970:
13967:
13965:
13962:
13960:
13957:
13955:
13952:
13950:
13947:
13945:
13942:
13940:
13937:
13935:
13932:
13930:
13927:
13925:
13922:
13921:
13919:
13915:
13909:
13906:
13904:
13901:
13898:
13895:
13893:
13890:
13888:
13885:
13883:
13880:
13878:
13875:
13873:
13870:
13868:
13865:
13863:
13860:
13858:
13855:
13854:
13852:
13848:
13842:
13839:
13837:
13834:
13832:
13829:
13827:
13824:
13823:
13821:
13817:
13814:
13810:
13800:
13797:
13793:
13790:
13789:
13788:
13785:
13782:
13779:
13778:
13776:
13772:
13766:
13763:
13761:
13758:
13756:
13753:
13751:
13748:
13746:
13743:
13741:
13738:
13736:
13733:
13731:
13728:
13726:
13723:
13721:
13718:
13717:
13715:
13711:
13705:
13702:
13700:
13697:
13695:
13692:
13690:
13687:
13685:
13682:
13680:
13677:
13675:
13672:
13670:
13667:
13665:
13662:
13660:
13657:
13656:
13654:
13650:
13647:
13643:
13639:
13632:
13627:
13625:
13620:
13618:
13613:
13612:
13609:
13603:
13600:
13597:
13594:
13591:
13587:
13584:
13581:
13580:Gerald Teschl
13577:
13574:
13571:
13568:
13565:
13562:
13559:
13555:
13552:
13549:
13546:
13542:
13538:
13537:
13532:
13528:
13527:
13518:
13514:
13512:
13511:0-415-27267-X
13508:
13504:
13500:
13497:
13493:
13487:
13483:
13479:
13475:
13474:
13469:
13465:
13460:
13458:0-8493-4488-3
13454:
13450:
13446:
13442:
13440:
13439:0-486-49510-8
13436:
13432:
13428:
13425:
13422:
13418:
13414:
13408:
13404:
13403:
13398:
13394:
13392:
13388:
13387:
13382:
13379:
13375:
13371:
13365:
13361:
13357:
13353:
13349:
13348:
13343:
13339:
13335:
13330:
13329:
13323:
13318:
13317:
13308:
13302:
13298:
13293:
13290:
13284:
13280:
13276:
13271:
13264:
13263:
13257:
13254:
13252:0-87901-432-6
13248:
13244:
13240:
13235:
13232:
13228:
13224:
13220:
13216:
13212:
13210:
13209:1-58488-297-2
13206:
13202:
13198:
13194:
13192:0-471-50728-8
13188:
13184:
13179:
13178:
13172:
13168:
13165:
13163:0-13-487538-9
13159:
13155:
13154:Prentice-Hall
13151:
13146:
13143:
13141:0-471-71716-9
13137:
13133:
13129:
13125:
13121:
13117:
13116:
13103:
13101:
13094:
13093:0-7135-1594-5
13090:
13084:
13075:
13073:
13063:
13057:
13056:0-471-83824-1
13053:
13047:
13045:
13043:
13041:
13039:
13037:
13035:
13033:
13031:
13029:
13020:
13015:, p. 13)
13014:
13009:
13000:
12992:
12986:
12981:
12974:
12969:
12961:
12957:
12953:
12949:
12945:
12941:
12934:
12927:
12922:
12916:, p. 78)
12915:
12910:
12902:
12896:
12891:
12883:
12877:
12873:
12866:
12858:
12853:, p. 12)
12852:
12847:
12841:, p. 12)
12840:
12835:
12828:
12823:
12817:, p. 47)
12816:
12815:Simmons (1972
12811:
12805:, p. 24)
12804:
12799:
12797:
12789:
12788:Simmons (1972
12784:
12777:
12772:
12765:
12760:
12758:
12750:
12745:
12739:, p. 78)
12738:
12733:
12726:
12725:Simmons (1972
12721:
12715:, p. 64)
12714:
12709:
12702:
12696:
12687:
12682:
12675:
12667:
12661:
12657:
12650:
12642:
12638:
12633:
12628:
12624:
12620:
12616:
12612:
12608:
12601:
12592:
12587:
12580:
12566:
12562:
12558:
12552:
12536:
12532:
12526:
12522:
12521:
12513:
12509:
12500:
12497:
12495:
12492:
12490:
12487:
12485:
12482:
12480:
12477:
12475:
12472:
12470:
12467:
12466:
12457:
12454:
12451:
12447:
12444:
12441:
12438:
12435:
12432:
12429:
12426:
12423:
12420:
12417:
12414:
12411:
12408:
12405:
12402:
12399:
12396:
12393:
12390:
12387:
12383:
12380:
12377:
12373:
12370:
12369:
12363:
12345:
12337:
12324:
12320:
12304:
12301:
12297:
12293:
12290:
12287:
12271:
12268:
12260:
12250:
12246:
12242:
12236:
12235:
12231:
12226:This section
12224:
12220:
12215:
12214:
12210:
12196:
12193:
12189:
12168:
12164:
12160:
12157:
12152:
12148:
12141:
12138:
12133:
12128:
12124:
12119:
12113:
12109:
12105:
12100:
12095:
12091:
12086:
12080:
12076:
12067:
12062:
12058:
12053:
12049:
12045:
12040:
12036:
12018:
12013:
12009:
12004:
11999:
11993:
11990:
11987:
11983:
11977:
11974:
11971:
11967:
11959:
11955:
11949:
11946:
11943:
11939:
11934:
11928:
11923:
11920:
11917:
11913:
11909:
11904:
11900:
11891:
11883:
11879:
11861:
11856:
11852:
11847:
11841:
11837:
11831:
11826:
11823:
11820:
11816:
11812:
11807:
11803:
11793:
11789:
11773:
11770:
11762:
11758:
11754:
11751:
11743:
11738:
11735:
11732:
11728:
11719:
11716:
11712:
11707:
11703:
11684:
11680:
11676:
11671:
11667:
11663:
11660:
11653:
11651:
11649:
11645:
11641:
11637:
11614:
11609:
11605:
11600:
11591:
11585:
11581:
11573:
11566:
11564:
11548:
11542:
11539:
11531:
11527:
11523:
11518:
11513:
11509:
11500:
11496:
11490:
11485:
11482:
11479:
11475:
11465:
11462:
11461:
11458:
11444:
11439:
11433:
11427:
11423:
11419:
11416:
11413:
11406:
11402:
11398:
11395:
11390:
11386:
11382:
11378:
11372:
11366:
11362:
11358:
11355:
11352:
11345:
11341:
11337:
11334:
11329:
11325:
11320:
11312:
11308:
11305:
11299:
11295:
11291:
11286:
11282:
11273:
11270:
11266:
11260:
11243:
11239:
11236:
11230:
11226:
11217:
11213:
11209:
11206:
11201:
11197:
11190:
11185:
11181:
11172:
11169:
11165:
11159:
11143:
11137:
11134:
11131:
11126:
11122:
11116:
11113:
11109:
11102:
11099:
11094:
11090:
11084:
11080:
11076:
11070:
11064:
11061:
11058:
11053:
11049:
11043:
11040:
11036:
11029:
11026:
11021:
11017:
11011:
11007:
11003:
10998:
10994:
10985:
10982:
10978:
10958:
10954:
10950:
10945:
10941:
10937:
10934:
10927:
10925:
10923:
10919:
10915:
10911:
10888:
10883:
10879:
10874:
10865:
10859:
10855:
10847:
10840:
10838:
10822:
10816:
10813:
10810:
10807:
10804:
10798:
10795:
10790:
10787:
10781:
10778:
10770:
10766:
10762:
10757:
10752:
10748:
10735:
10734:
10718:
10712:
10708:
10704:
10701:
10696:
10692:
10688:
10685:
10682:
10678:
10674:
10671:
10664:
10658:
10653:
10650:
10643:
10637:
10632:
10628:
10623:
10617:
10613:
10608:
10602:
10598:
10593:
10587:
10584:
10577:
10571:
10566:
10562:
10558:
10554:
10550:
10547:
10540:
10524:
10521:
10514:
10508:
10503:
10499:
10494:
10485:
10483:
10467:
10461:
10458:
10455:
10451:
10444:
10437:
10434:
10430:
10425:
10417:
10411:
10407:
10403:
10397:
10394:
10389:
10386:
10377:
10371:
10368:
10365:
10357:
10353:
10349:
10344:
10339:
10335:
10322:
10321:
10318:
10304:
10300:
10297:
10294:
10291:
10284:
10278:
10273:
10270:
10263:
10257:
10252:
10248:
10243:
10237:
10233:
10228:
10222:
10219:
10212:
10206:
10201:
10197:
10193:
10189:
10185:
10182:
10173:
10159:
10154:
10151:
10144:
10138:
10133:
10129:
10124:
10115:
10113:
10097:
10091:
10088:
10085:
10079:
10073:
10070:
10064:
10061:
10056:
10053:
10041:
10040:
10029:
10023:
9999:
9995:
9991:
9983:
9979:
9975:
9972:
9969:
9962:
9956:
9951:
9947:
9943:
9938:
9935:
9927:
9923:
9919:
9916:
9913:
9906:
9887:
9882:
9876:
9873:
9868:
9865:
9859:
9851:
9848:
9844:
9839:
9831:
9827:
9823:
9818:
9813:
9809:
9799:
9796:
9791:
9788:
9782:
9774:, substitute
9772:
9768:
9762:
9760:
9744:
9738:
9735:
9727:
9723:
9719:
9714:
9709:
9705:
9692:
9691:
9680:
9668:
9655:
9652:
9645:
9642:
9639:
9633:
9627:
9624:
9621:
9615:
9610:
9606:
9599:
9587:
9581:
9578:
9575:
9569:
9566:
9560:
9554:
9534:
9531:
9524:
9521:
9518:
9512:
9506:
9503:
9500:
9494:
9489:
9485:
9478:
9466:
9460:
9457:
9454:
9448:
9445:
9439:
9433:
9424:
9407:
9404:
9401:
9398:
9391:
9385:
9380:
9376:
9372:
9369:
9366:
9359:
9356:
9353:
9347:
9341:
9338:
9335:
9329:
9324:
9320:
9317:
9312:
9305:
9302:
9295:
9289:
9284:
9280:
9276:
9273:
9270:
9263:
9260:
9257:
9251:
9245:
9242:
9239:
9233:
9228:
9224:
9221:
9216:
9210:
9207:
9204:
9198:
9183:
9179:
9175:
9170:
9168:
9152:
9141:
9138:
9126:
9120:
9109:
9106:
9086:
9082:
9078:
9074:
9071:
9069:
9053:
9045:
9036:
9030:
9022:
9007:
8990:
8987:
8985:
8980:
8977:
8970:
8967:
8964:
8958:
8955:
8952:
8949:
8942:
8939:
8936:
8930:
8923:
8920:
8918:
8910:
8907:
8904:
8898:
8895:
8889:
8886:
8881:
8878:
8869:
8866:
8863:
8857:
8844:
8841:
8840:
8837:
8824:
8821:
8814:
8811:
8808:
8802:
8797:
8793:
8786:
8774:
8768:
8765:
8762:
8756:
8753:
8747:
8741:
8721:
8718:
8711:
8708:
8705:
8699:
8694:
8690:
8683:
8671:
8665:
8662:
8659:
8653:
8650:
8644:
8638:
8613:
8610:
8607:
8604:
8597:
8591:
8586:
8582:
8578:
8575:
8572:
8565:
8562:
8559:
8553:
8548:
8544:
8540:
8538:
8530:
8527:
8520:
8514:
8509:
8505:
8501:
8498:
8495:
8488:
8485:
8482:
8476:
8471:
8467:
8463:
8461:
8453:
8450:
8447:
8441:
8430:
8427:
8425:
8409:
8401:
8392:
8386:
8378:
8363:
8346:
8343:
8341:
8336:
8333:
8326:
8323:
8320:
8314:
8311:
8308:
8305:
8298:
8295:
8292:
8286:
8279:
8276:
8274:
8266:
8263:
8260:
8254:
8251:
8245:
8242:
8237:
8234:
8225:
8222:
8219:
8213:
8200:
8197:
8196:
8193:
8191:
8187:
8183:
8179:
8174:
8152:
8146:
8143:
8137:
8131:
8125:
8120:
8117:
8110:
8104:
8096:
8093:
8089:
8085:
8079:
8076:
8070:
8067:
8059:
8056:
8052:
8050:
8033:
8030:
8028:
8023:
8020:
8013:
8010:
8004:
8001:
7998:
7995:
7992:
7985:
7982:
7976:
7973:
7966:
7963:
7961:
7953:
7950:
7945:
7942:
7932:
7929:
7923:
7920:
7917:
7911:
7908:
7902:
7899:
7886:
7885:
7867:
7864:
7858:
7852:
7847:
7844:
7836:
7832:
7828:
7824:
7820:
7814:
7811:
7805:
7802:
7795:
7792:
7788:
7784:
7780:
7778:
7764:
7759:
7756:
7751:
7747:
7744:
7738:
7735:
7730:
7727:
7715:
7714:
7703:
7679:
7676:
7673:
7670:
7663:
7657:
7652:
7648:
7644:
7641:
7638:
7631:
7625:
7620:
7616:
7608:
7605:
7603:
7586:
7583:
7581:
7576:
7573:
7566:
7560:
7557:
7554:
7551:
7544:
7538:
7531:
7528:
7526:
7518:
7512:
7509:
7503:
7500:
7495:
7492:
7483:
7477:
7465:
7461:
7457:
7456:
7441:
7438:
7429:
7423:
7418:
7415:
7407:
7403:
7399:
7396:
7389:
7386:
7382:
7379:
7377:
7360:
7357:
7350:
7344:
7341:
7339:
7334:
7331:
7321:
7315:
7312:
7310:
7302:
7299:
7294:
7291:
7276:
7272:
7271:
7256:
7253:
7250:
7247:
7240:
7234:
7229:
7225:
7221:
7218:
7211:
7208:
7206:
7189:
7186:
7179:
7173:
7170:
7168:
7163:
7160:
7150:
7144:
7141:
7139:
7131:
7128:
7123:
7120:
7105:
7101:
7100:
7085:
7082:
7079:
7076:
7066:
7058:
7054:
7045:
7037:
7033:
7024:
7020:
7016:
7013:
7010:
7000:
6992:
6988:
6979:
6971:
6967:
6958:
6954:
6946:
6940:
6934:
6930:
6928:
6911:
6908:
6906:
6901:
6898:
6891:
6883:
6879:
6872:
6864:
6860:
6856:
6853:
6850:
6843:
6835:
6831:
6824:
6816:
6812:
6804:
6801:
6799:
6791:
6788:
6783:
6780:
6770:
6762:
6758:
6751:
6743:
6739:
6735:
6729:
6721:
6717:
6710:
6702:
6698:
6684:
6680:
6676:
6675:
6664:
6656:
6653:
6649:
6644:
6634:
6630:
6624:
6620:
6616:
6610:
6605:
6603:
6594:
6587:
6581:
6575:
6569:
6563:
6559:functions of
6558:
6552:
6548:
6544:
6537:
6533:
6529:
6522:
6518:
6511:
6507:
6500:
6496:
6489:
6485:
6479:
6457:
6446:
6442:
6438:
6435:
6432:
6427:
6423:
6419:
6416:
6410:
6407:
6402:
6398:
6394:
6391:
6388:
6383:
6379:
6372:
6364:
6360:
6356:
6353:
6350:
6345:
6341:
6337:
6334:
6322:
6314:
6310:
6306:
6303:
6300:
6295:
6291:
6284:
6272:
6271:
6270:
6245:
6242:
6231:
6227:
6214:
6213:
6212:
6191:
6183:
6179:
6175:
6172:
6169:
6164:
6160:
6156:
6153:
6147:
6142:
6136:
6132:
6128:
6118:
6114:
6108:
6102:
6098:
6095:
6092:
6088:
6079:
6069:
6065:
6059:
6053:
6049:
6045:
6035:
6031:
6025:
6019:
6015:
6011:
5999:
5998:
5997:
5995:
5991:
5988:= 1, 2, ...,
5987:
5967:
5959:
5956:
5953:
5946:
5942:
5937:
5933:
5925:
5924:
5923:
5921:
5897:
5890:
5886:
5882:
5873:
5870:
5867:
5860:
5853:
5850:
5844:
5840:
5837:
5833:
5829:
5826:
5822:
5819:
5816:
5813:
5809:
5805:
5798:
5797:
5796:
5794:
5784:
5782:
5772:
5755:
5740:
5734:
5724:
5720:
5713:
5701:
5700:
5699:
5674:
5659:
5651:
5647:
5642:
5638:
5635:
5630:
5626:
5606:
5580:
5577:
5572:
5568:
5561:
5554:
5551:
5544:
5540:
5536:
5531:
5526:
5522:
5517:
5509:
5506:
5501:
5497:
5490:
5482:
5478:
5474:
5469:
5464:
5460:
5456:
5450:
5446:
5438:
5435:
5430:
5426:
5412:
5403:
5402:
5401:
5381:
5378:
5373:
5369:
5362:
5359:
5354:
5350:
5340:
5336:
5330:
5324:
5318:
5311:
5310:
5309:
5284:
5282:
5278:
5274:
5270:
5249:
5245:
5241:
5237:
5234:
5226:
5225:
5221:
5220:
5213:
5206:
5202:
5178:
5175:
5174:
5173:
5170:
5136:
5102:
5093:
5087:
5080:
5076:
5069:
5056:
5030:
5024:
5014:
5010:
5003:
4990:
4989:
4988:
4971:
4968:
4963:
4959:
4949:
4929:
4920:
4895:
4891:
4886:
4882:
4878:
4869:
4863:
4855:
4850:
4846:
4842:
4834:
4830:
4826:
4821:
4817:
4810:
4801:
4793:
4792:
4791:
4786:
4779:
4774:
4766:
4764:
4760:
4756:
4752:
4748:
4743:
4739:
4719:
4716:
4711:
4707:
4703:
4700:
4697:
4692:
4688:
4681:
4675:
4672:
4667:
4663:
4659:
4656:
4653:
4648:
4644:
4637:
4634:
4626:
4622:
4613:
4609:
4605:
4600:
4596:
4592:
4589:plane, where
4588:
4569:
4566:
4561:
4557:
4553:
4550:
4547:
4542:
4538:
4531:
4525:
4522:
4517:
4513:
4509:
4506:
4503:
4498:
4494:
4487:
4484:
4476:
4472:
4468:
4447:
4443:
4436:
4433:
4428:
4424:
4419:
4412:
4409:
4406:
4400:
4397:
4393:
4390:
4375:
4373:
4368:
4366:
4354:
4352:
4348:
4346:
4343:
4342:
4338:
4336:
4332:
4330:
4327:
4326:
4322:
4319:
4316:
4315:
4312:
4311:
4310:
4308:
4298:
4296:
4292:
4288:
4284:
4280:
4274:
4264:
4261:
4259:
4255:
4251:
4247:
4243:
4239:
4235:
4230:
4226:
4224:
4220:
4216:
4212:
4202:
4200:
4196:
4192:
4188:
4184:
4178:
4168:
4165:
4161:
4157:
4153:
4149:
4139:
4137:
4133:
4129:
4124:
4120:
4089:
4084:
4079:
4073:
4070:
4065:
4062:
4058:
4054:
4049:
4046:
4041:
4038:
4034:
4027:
4024:
4019:
4013:
4007:
4000:
3999:
3998:
3981:
3978:
3972:
3966:
3961:
3951:
3938:
3927:
3924:
3920:
3917:
3909:
3908:
3907:
3904:
3895:
3893:
3889:
3885:
3881:
3877:
3872:
3870:
3866:
3862:
3858:
3854:
3850:
3846:
3842:
3837:
3835:
3831:
3827:
3807:
3804:
3801:
3798:
3791:
3785:
3782:
3776:
3770:
3763:
3762:
3761:
3758:
3754:
3749:
3745:
3742:is called an
3741:
3736:
3732:
3728:
3724:
3718:
3714:
3710:
3706:
3686:
3683:
3680:
3677:
3673:
3670:
3659:
3652:
3645:
3642:
3636:
3632:
3629:
3625:
3622:
3619:
3616:
3610:
3603:
3602:
3601:
3599:
3595:
3591:
3587:
3583:
3579:
3575:
3570:
3566:
3562:
3558:
3538:
3535:
3531:
3522:
3515:
3511:
3508:
3505:
3501:
3498:
3494:
3491:
3488:
3485:
3481:
3477:
3470:
3469:
3468:
3460:
3458:
3453:
3451:
3447:
3443:
3438:
3401:
3393:
3390:
3369:
3348:
3344:
3339:
3330:
3322:
3319:
3315:
3284:
3278:
3271:
3264:
3257:
3251:
3246:
3241:
3227:
3214:
3211:
3208:
3204:
3195:
3191:
3182:
3174:
3171:
3163:
3159:
3151:
3136:
3123:
3120:
3117:
3113:
3104:
3100:
3091:
3083:
3080:
3072:
3068:
3052:
3039:
3036:
3033:
3029:
3020:
3016:
3007:
2999:
2996:
2988:
2984:
2977:
2968:
2967:
2966:
2964:
2960:
2936:
2932:
2923:
2910:
2907:
2904:
2900:
2891:
2887:
2878:
2870:
2867:
2863:
2850:
2849:
2848:
2847:analogue is:
2846:
2825:
2818:
2809:
2806:
2803:
2790:
2787:
2784:
2780:
2771:
2767:
2758:
2750:
2747:
2743:
2737:
2733:
2725:
2717:
2708:
2705:
2702:
2689:
2686:
2683:
2679:
2670:
2666:
2657:
2649:
2646:
2642:
2636:
2632:
2623:
2614:
2611:
2608:
2595:
2592:
2589:
2585:
2576:
2572:
2563:
2555:
2552:
2548:
2542:
2538:
2531:
2526:
2521:
2510:
2502:
2498:
2490:
2478:
2470:
2466:
2453:
2445:
2441:
2434:
2425:
2424:
2423:
2421:
2420:column vector
2417:
2414:
2410:
2407:
2403:
2383:
2374:
2371:
2368:
2355:
2352:
2349:
2345:
2336:
2332:
2323:
2315:
2312:
2308:
2299:
2291:
2274:
2273:
2272:
2270:
2266:
2262:
2258:
2254:
2250:
2244:
2231:
2228:
2218:
2214:
2209:
2205:
2202:
2193:
2188:
2182:
2178:
2173:
2169:
2166:
2161:
2158:
2154:
2150:
2140:
2136:
2129:
2121:
2116:
2094:
2088:
2085:
2077:
2070:
2063:
2055:
2051:
2045:
2042:
2039:
2034:
2031:
2028:
2024:
2020:
2012:
2005:
1997:
1996:
1993:
1988:
1972:
1964:
1960:
1957:
1949:
1944:
1940:
1937:
1932:
1931:
1914:
1911:
1907:
1898:
1891:
1884:
1881:
1875:
1871:
1868:
1864:
1860:
1857:
1853:
1850:
1847:
1844:
1840:
1836:
1829:
1828:
1827:
1825:
1821:
1816:
1814:
1812:
1808:is called an
1787:
1780:
1776:
1772:
1763:
1760:
1757:
1750:
1746:
1743:
1740:
1736:
1733:
1729:
1726:
1723:
1720:
1716:
1712:
1705:
1704:
1703:
1701:
1697:
1693:
1689:
1683:
1673:
1653:
1650:
1647:
1643:
1638:
1632:
1629:
1623:
1617:
1614:
1601:
1597:
1591:
1587:
1583:
1578:
1574:
1564:
1556:
1553:
1541:
1533:
1530:
1518:
1510:
1501:
1497:
1493:
1490:
1484:
1480:
1476:
1471:
1467:
1457:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1389:
1383:
1377:
1374:
1366:
1362:
1348:
1342:
1337:
1324:
1317:
1316:
1315:
1313:
1309:
1306:and the time
1305:
1301:
1296:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1265:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1227:
1222:
1220:
1216:
1212:
1208:
1198:
1196:
1192:
1191:Taylor series
1188:
1183:
1181:
1177:
1173:
1169:
1165:
1161:
1156:
1127:
1120:
1116:
1113:
1110:
1106:
1103:
1092:
1071:
1065:
1038:
1030:
1026:
998:
990:
986:
960:
957:
954:
948:
942:
939:
931:
924:
917:
909:
905:
901:
898:
895:
891:
888:
881:
873:
869:
865:
861:
858:
851:
843:
839:
835:
832:
826:
818:
814:
806:
805:
804:
803:of the form
802:
798:
794:
784:
782:
780:
773:
771:
769:
764:
760:
756:
752:
748:
744:
732:
727:
725:
720:
718:
713:
712:
710:
709:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
622:
615:
614:
610:
609:
602:
599:
597:
594:
592:
589:
588:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
557:
554:
553:
552:
549:
547:
546:Finite volume
544:
540:
537:
536:
535:
532:
528:
522:
519:
517:
514:
512:
508:
506:
503:
500:
499:
492:
491:
483:
480:
478:
475:
471:
467:
465:
462:
460:
456:
452:
449:
447:
444:
442:
439:
435:
432:
430:
427:
425:
422:
420:
417:
416:
415:
412:
410:
407:
406:
399:
398:
391:
388:
386:
383:
381:
378:
376:
373:
372:
366:
365:
361:
360:
352:
349:
345:
342:
341:
340:
337:
336:
332:
326:
325:
319:
318:
307:
304:
303:
299:
296:
294:
291:
290:
288:
287:
283:
282:
276:
272:
269:
267:
264:
261:
259:
256:
255:
251:
248:
247:
246:
245:
241:
240:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
200:
198:
197:
189:
188:
184:
183:
178:
168:
165:
163:
160:
159:
158:
157:
154:
151:
150:
145:
142:
140:
137:
135:
132:
131:
130:
129:
126:
123:
122:
117:
114:
112:
108:
106:
103:
101:
98:
96:
93:
92:
91:
90:
84:
81:
79:
76:
75:
73:
72:
64:
63:
59:
58:
55:
52:
51:
45:
41:
37:
32:
19:
14347:
14189:
14168:
14110:Martin Kutta
14065:Émile Picard
14045:Isaac Newton
13959:Euler method
13929:Substitution
13668:
13534:
13516:
13502:
13472:
13448:
13430:
13401:
13385:
13383:W. Johnson,
13346:
13332:. New York:
13327:
13314:Bibliography
13296:
13274:
13261:
13238:
13221:, New York:
13218:
13200:
13176:
13149:
13127:
13083:
13062:
13013:Ascher (1998
13008:
12999:
12987:, p. 9)
12980:
12975:, 1866, 1868
12972:
12968:
12943:
12933:
12928:, p. 4)
12921:
12909:
12897:, p. 5)
12895:Ascher (1998
12890:
12871:
12865:
12851:Ascher (1998
12846:
12834:
12827:Harper (1976
12822:
12810:
12790:, p. 3)
12783:
12778:, p. 2)
12771:
12764:Harper (1976
12749:Tipler (1991
12744:
12732:
12720:
12708:
12695:
12674:
12655:
12649:
12614:
12610:
12600:
12579:
12568:. Retrieved
12560:
12551:
12539:. Retrieved
12519:
12512:
12325:
12321:
12278:
12263:
12257:January 2020
12254:
12239:Please help
12227:
12191:
12187:
12064:, and using
12060:
12056:
12051:
12047:
12043:
12038:
12034:
11886:
11881:
11877:
11791:
11787:
11786:, then: for
11717:
11705:
11701:
11699:
11647:
11643:
11632:
11630:
11583:
11576:
11568:
11467:
11463:
11274:
11268:
11264:
11261:
11173:
11167:
11163:
11160:
10986:
10980:
10976:
10973:
10921:
10917:
10906:
10904:
10857:
10850:
10842:
10737:
10324:
10175:
10043:
10021:
9770:
9766:
9694:
9425:
9187:
9181:
9177:
9173:
9090:
9084:
9080:
9076:
9008:
8846:
8630:
8364:
8202:
8189:
8185:
8181:
8177:
8175:
8060:
8054:
7888:
7790:
7786:
7782:
7717:
7466:
7463:
7459:
7384:
7277:
7274:
7106:
7103:
6938:
6932:
6686:
6682:
6678:
6651:
6647:
6642:
6632:
6628:
6622:
6618:
6614:
6606:
6592:
6585:
6579:
6573:
6567:
6561:
6550:
6546:
6542:
6535:
6531:
6527:
6520:
6516:
6509:
6505:
6498:
6494:
6487:
6483:
6480:
6477:
6268:
6210:
5993:
5989:
5985:
5983:
5919:
5917:
5792:
5790:
5778:
5770:
5675:
5604:
5602:
5399:
5285:
5280:
5276:
5272:
5268:
5266:
5211:
5204:
5171:
5134:
5132:
4950:
4916:
4784:
4777:
4775:
4772:
4762:
4758:
4754:
4750:
4746:
4741:
4737:
4619:denotes the
4611:
4607:
4603:
4594:
4590:
4586:
4474:
4470:
4466:
4381:
4369:
4362:
4304:
4276:
4262:
4246:Lie algebras
4242:group theory
4231:
4227:
4208:
4205:Lie's theory
4198:
4180:
4151:
4145:
4116:
3996:
3905:
3901:
3879:
3875:
3873:
3856:
3848:
3844:
3840:
3838:
3833:
3832:is called a
3829:
3825:
3823:
3756:
3752:
3747:
3743:
3739:
3734:
3730:
3726:
3722:
3716:
3712:
3708:
3704:
3701:
3597:
3593:
3589:
3585:
3577:
3573:
3568:
3564:
3560:
3556:
3553:
3466:
3454:
3437:non-singular
3301:
2958:
2956:
2844:
2842:
2415:
2412:
2408:
2405:
2401:
2399:
2268:
2260:
2256:
2252:
2248:
2246:
2216:
2212:
2207:
2191:
2180:
2176:
2171:
2156:
2152:
2148:
2138:
2134:
2127:
2119:
2114:
1991:
1962:
1947:
1942:
1929:
1823:
1819:
1817:
1809:
1807:
1699:
1695:
1691:
1687:
1685:
1589:
1585:
1581:
1562:
1554:
1551:
1539:
1531:
1528:
1516:
1508:
1491:
1482:
1478:
1474:
1465:
1463:
1453:
1449:
1445:
1441:
1437:
1433:
1429:
1425:
1421:
1417:
1411:
1311:
1307:
1303:
1297:
1266:
1223:
1204:
1184:
1157:
975:
790:
778:
767:
750:
746:
740:
690:Émile Picard
675:Martin Kutta
665:George Green
625:Isaac Newton
457: /
453: /
273: /
202:
139:Chaos theory
14175:Integration
13867:Phase space
13725:Homogeneous
13334:McGraw-Hill
13268:(in French)
13223:McGraw-Hill
12617:(1): 1–29.
12416:Mathematica
9089:satisfying
7383:(divide by
5279:, which is
4919:restriction
4323:Conclusion
4320:Assumption
4279:eigenvalues
4209:From 1870,
4148:quadratures
3880:homogeneous
3554:a function
2963:zero vector
2172:homogeneous
2165:Homogeneous
2157:source term
1573:integration
1460:Definitions
1242:meteorology
1207:mathematics
763:derivatives
743:mathematics
583:Runge–Kutta
328:Difference
271:Homogeneous
83:Engineering
14363:Categories
14200:stochastic
14095:John Crank
13924:Inspection
13787:Stochastic
13781:Difference
13755:Autonomous
13699:Non-linear
13689:Fractional
13652:Operations
13478:Providence
13113:References
12686:2012.07558
12591:2206.00364
12570:2016-07-28
12398:GNU Octave
12384:, a free (
11711:polynomial
11592:functions
10866:functions
6557:integrable
4335:continuous
4238:Lie theory
4215:Lie groups
4211:Sophus Lie
2259:) = , and
2227:Non-linear
1943:autonomous
1936:Autonomous
1680:See also:
1575:, whereas
1289:d'Alembert
1201:Background
1160:elementary
779:stochastic
700:John Crank
501:Inspection
455:Asymptotic
339:Stochastic
258:Autonomous
233:Non-linear
223:Fractional
40:projectile
36:trajectory
14312:Functions
13899:solutions
13857:Wronskian
13812:Solutions
13740:Decoupled
13704:Holonomic
13541:EMS Press
13399:(1944) ,
13344:(2002) ,
12641:0377-0427
12302:α
12228:does not
12165:φ
12149:γ
12142:
12125:χ
12092:α
12033:for some
12010:α
11991:−
11988:ℓ
11978:ℓ
11944:ℓ
11940:∑
11914:∑
11885:repeated
11853:α
11817:∑
11759:α
11755:−
11752:α
11729:∏
11606:α
11574:: assume
11476:∑
11420:−
11399:
11359:−
11338:
11300:−
11231:−
11132:−
11117:−
11095:−
11059:−
11022:−
10880:α
10848:: assume
10686:ξ
10675:λ
10665:λ
10654:ε
10644:ε
10633:λ
10629:∫
10618:ξ
10614:∫
10599:∫
10588:λ
10578:λ
10563:∫
10559:−
10525:λ
10515:λ
10500:∫
10295:λ
10285:λ
10274:ε
10264:ε
10253:λ
10249:∫
10234:∫
10223:λ
10213:λ
10198:∫
10194:−
10155:λ
10145:λ
10130:∫
9973:ε
9963:ε
9952:λ
9948:∫
9939:λ
9924:∫
9920:±
9656:λ
9646:λ
9628:λ
9616:μ
9607:∫
9597:∂
9593:∂
9588:−
9535:λ
9519:λ
9501:λ
9495:μ
9486:∫
9476:∂
9472:∂
9467:−
9402:λ
9392:λ
9377:∫
9370:λ
9360:λ
9342:λ
9330:μ
9321:∫
9306:λ
9296:λ
9281:∫
9274:λ
9258:λ
9240:λ
9234:μ
9225:∫
9150:∂
9139:μ
9133:∂
9118:∂
9107:μ
9101:∂
9051:∂
9043:∂
9037:≠
9028:∂
9020:∂
8825:λ
8815:λ
8794:∫
8784:∂
8780:∂
8775:−
8722:λ
8706:λ
8691:∫
8681:∂
8677:∂
8672:−
8608:λ
8598:λ
8583:∫
8576:λ
8566:λ
8545:∫
8531:λ
8521:λ
8506:∫
8499:λ
8483:λ
8468:∫
8407:∂
8399:∂
8384:∂
8376:∂
8153:λ
8144:−
8138:λ
8126:λ
8121:λ
8111:λ
8090:∫
8071:
7868:λ
7865:−
7859:λ
7848:λ
7825:∫
7806:
7674:λ
7664:λ
7649:∫
7642:λ
7632:λ
7617:∫
7430:λ
7419:λ
7404:∫
7251:λ
7241:λ
7226:∫
7080:λ
7067:λ
7046:λ
7021:∫
7014:λ
7001:λ
6980:λ
6955:∫
6609:summation
6436:…
6392:…
6354:…
6304:…
6173:…
6096:−
6080:⋮
5957:−
5871:−
5851:…
5753:∞
5750:→
5747:‖
5732:‖
5725:±
5717:→
5620:∖
5558:∞
5454:∞
5451:−
5360:−
5154:¯
5151:Ω
5145:∂
5115:¯
5112:Ω
5106:∂
5103:∈
5081:±
5073:→
5043:∞
5040:→
5037:‖
5022:‖
5015:±
5007:→
4975:∞
4972:±
4969:≠
4964:±
4892:∈
4873:∞
4870:±
4864:∪
4856:∈
4851:±
4822:−
4735:for some
4698:−
4682:⊂
4654:−
4548:−
4532:×
4504:−
4367:are met.
4254:Lax pairs
4187:Frobenius
4066:−
4042:−
3928:−
3802:∈
3744:extension
3681:∈
3643:…
3509:…
3463:Solutions
3415:∂
3380:∂
3272:⋮
3212:…
3152:⋮
3121:…
3037:…
2908:…
2807:−
2788:…
2726:⋮
2706:−
2687:…
2612:−
2593:…
2491:⋮
2413:dimension
2372:−
2353:…
2043:−
2025:∑
1882:…
1761:−
1744:…
1633:¨
1618:˙
1262:economics
1246:chemistry
1238:astronomy
1219:gradients
1187:integrals
1114:…
899:⋯
774:more than
441:Wronskian
419:Dirichlet
162:Economics
105:Chemistry
95:Astronomy
14337:Infinity
14190:ordinary
14170:Calculus
14007:Examples
13897:Integral
13669:Ordinary
13470:(2012).
13447:(1993).
13324:(1955).
13299:, SIAM,
13231:75173716
13217:(1972),
13173:(1972),
13126:(1977),
12960:45426376
12535:Archived
12463:See also
12434:SageMath
10438:′
6555:are any
6228:′
6137:′
6103:′
6054:′
6020:′
5841:″
5830:′
5698:because
5238:′
5222:Example.
4394:′
4317:Theorem
4132:Casorati
4108:Theories
3921:′
3633:′
3578:solution
3572:, where
3502:′
3340:′
3205:″
3192:′
3114:″
3101:′
3030:″
3017:′
2901:″
2888:′
2845:implicit
2781:″
2768:′
2680:″
2667:′
2586:″
2573:′
2346:″
2333:′
2137: (
1872:″
1861:′
1820:implicit
1811:explicit
1737:′
1285:Clairaut
1226:geometry
1107:′
892:″
862:′
801:equation
759:variable
551:Galerkin
451:Lyapunov
362:Solution
306:Notation
298:Operator
284:Features
203:Ordinary
14195:partial
13735:Coupled
13674:Partial
13543:, 2001
13421:0010757
13378:1929104
13128:Physics
12619:Bibcode
12541:11 July
12446:Chebfun
12249:removed
12234:sources
11892:times,
11272:, then
11171:, then
10984:, then
6641:, then
6602:complex
4585:in the
4191:Clebsch
4128:Darboux
2124:
2117:
1588:′′, …,
1567:
1548:
1544:
1525:
1521:
1505:
1487:of the
1281:Riccati
1273:Leibniz
1254:ecology
1250:biology
1234:physics
1168:physics
1164:special
1145:
1095:
1087:
1058:
1054:
1018:
1016:, ...,
1014:
978:
768:partial
753:) is a
424:Neumann
208:Partial
116:Geology
111:Biology
100:Physics
14332:Series
13750:Degree
13694:Linear
13509:
13488:
13455:
13437:
13419:
13409:
13376:
13366:
13303:
13285:
13249:
13229:
13207:
13189:
13160:
13138:
13091:
13054:
12973:Crelle
12958:
12878:
12662:
12639:
12527:
12450:MATLAB
12404:Scilab
12392:MATLAB
12382:COPASI
12372:Maxima
12186:where
11715:degree
11700:Since
11267:< 4
10979:> 4
9426:where
9009:where
8631:where
8365:where
7783:y = ux
6525:, and
6269:where
5992:. The
5857:
5848:
5100:
4745:where
4615:) and
4248:, and
4164:Cauchy
4136:Cayley
3867:'. A
3843:of an
3649:
3640:
3600:, and
2957:where
2422:form:
2400:is an
2112:where
1963:linear
1956:Linear
1888:
1879:
1686:Given
1494:. The
1291:, and
1275:, the
1269:Newton
1211:social
976:where
611:People
523:
470:Series
228:Linear
67:Fields
44:cannon
14327:Limit
13799:Delay
13745:Order
13266:(PDF)
13183:Wiley
13132:Wiley
12956:S2CID
12681:arXiv
12586:arXiv
12505:Notes
12456:GNU R
12440:SciPy
12422:SymPy
12410:Maple
6643:after
6598:, ...
5781:order
4469:and ∂
4183:Fuchs
4156:Gauss
3855:. A
3588:, if
2418:. In
2406:order
2263:is a
2219:) ≠ 0
2183:) = 0
1546:, …,
1468:is a
1293:Euler
745:, an
511:Euler
429:Robin
351:Delay
293:Order
266:Exact
192:Types
60:Scope
38:of a
13590:UIUC
13507:ISBN
13486:ISBN
13453:ISBN
13435:ISBN
13407:ISBN
13364:ISBN
13301:ISBN
13283:ISBN
13247:ISBN
13227:LCCN
13205:ISBN
13187:ISBN
13158:ISBN
13136:ISBN
13089:ISBN
13052:ISBN
13019:help
12991:help
12901:help
12876:ISBN
12857:help
12660:ISBN
12637:ISSN
12543:2019
12525:ISBN
12232:any
12230:cite
9547:and
8734:and
7789:and
7781:Set
7462:and
6681:and
6577:and
5984:for
5578:<
5507:>
5275:) =
5273:x, y
4599:real
4597:are
4593:and
4293:and
4134:and
3890:and
3760:and
3720:and
3584:for
2411:and
2132:and
1456:)).
1256:and
1236:and
1228:and
1213:and
1209:and
1170:and
1162:and
1056:and
618:List
34:The
13356:doi
12948:doi
12627:doi
12615:125
12243:by
12139:cos
11713:of
11396:cos
11335:sin
11262:If
11166:= 4
11161:If
10974:If
9171:If
8176:If
8057:).
7387:).
6944:).
5710:lim
5066:lim
4934:max
4900:max
4806:max
4747:the
4587:x-y
4465:if
4372:DAE
3932:sgn
3863:or
3750:if
3746:of
3592:is
3580:or
3435:be
2404:of
2267:of
2210:if
2194:= 0
2174:if
1965:if
1584:′,
1283:,
1182:).
751:ODE
741:In
14365::
14173::
13539:,
13533:,
13484:.
13480::
13476:.
13429:,
13417:MR
13415:,
13374:MR
13372:,
13362:,
13354:,
13281:,
13245:,
13225:,
13185:,
13156:,
13134:,
13122:;
13099:^
13071:^
13027:^
12954:.
12942:.
12795:^
12756:^
12635:.
12625:.
12609:.
12563:.
12559:.
12533:.
12057:iγ
12055:+
12046:=
11720::
11629:.
11582:=
10903:.
10856:=
9771:dx
9767:dy
9180:,
9083:,
8192:.
8188:=
8186:xy
8180:=
8068:ln
8055:xy
7803:ln
7793:.
6650:=
6623:dλ
6621:)
6613:∫
6591:,
6571:;
6565:,
6540:,
6514:,
6503:,
6492:,
5795:,
5217:).
5210:,
4948:.
4783:,
4761:,
4740:∈
4610:∈
4606:,
4473:/∂
4349:F
4333:F
4244:,
4225:.
3894:.
3839:A
3836:.
3755:⊂
3738:,
3733:→
3729:⊂
3725::
3715:→
3711:⊂
3707::
3567:→
3563:⊂
3559::
3459:.
2221:.
2189:"
1815:.
1694:,
1563:dx
1540:dx
1523:,
1517:dx
1509:dy
1477:=
1295:.
1287:,
1279:,
1271:,
1155:.
791:A
14296:)
14292:(
14266:)
14262:(
14149:e
14142:t
14135:v
13630:e
13623:t
13616:v
13592:.
13582:.
13560:.
13494:.
13463:.
13461:.
13358::
13336:.
13196:.
13021:)
12993:)
12962:.
12950::
12903:)
12884:.
12859:)
12689:.
12683::
12668:.
12643:.
12629::
12621::
12594:.
12588::
12573:.
12545:.
12378:.
12346:+
12338:=
12305:t
12298:e
12294:A
12291:=
12288:y
12270:)
12264:(
12259:)
12255:(
12251:.
12237:.
12192:j
12188:ϕ
12174:)
12169:j
12161:+
12158:x
12153:j
12145:(
12134:x
12129:j
12120:e
12114:j
12110:C
12106:=
12101:x
12096:j
12087:e
12081:j
12077:C
12061:j
12052:j
12048:χ
12044:α
12039:j
12035:α
12019:x
12014:j
12005:e
12000:)
11994:1
11984:x
11975:,
11972:j
11968:C
11960:j
11956:k
11950:1
11947:=
11935:(
11929:n
11924:1
11921:=
11918:j
11910:=
11905:c
11901:y
11889:j
11887:k
11882:j
11878:α
11862:x
11857:j
11848:e
11842:j
11838:C
11832:n
11827:1
11824:=
11821:j
11813:=
11808:c
11804:y
11792:j
11788:α
11774:0
11771:=
11768:)
11763:j
11749:(
11744:n
11739:1
11736:=
11733:j
11718:n
11706:j
11702:α
11685:p
11681:y
11677:+
11672:c
11668:y
11664:=
11661:y
11648:x
11646:(
11644:r
11635:p
11633:y
11615:x
11610:j
11601:e
11586:,
11584:e
11579:c
11577:y
11571:c
11569:y
11552:)
11549:x
11546:(
11543:r
11540:=
11532:j
11528:x
11524:d
11519:y
11514:j
11510:d
11501:j
11497:b
11491:n
11486:0
11483:=
11480:j
11464:n
11445:]
11440:)
11434:2
11428:2
11424:b
11417:c
11414:4
11407:x
11403:(
11391:2
11387:C
11383:+
11379:)
11373:2
11367:2
11363:b
11356:c
11353:4
11346:x
11342:(
11330:1
11326:C
11321:[
11313:2
11309:x
11306:b
11296:e
11292:=
11287:c
11283:y
11269:c
11265:b
11244:2
11240:x
11237:b
11227:e
11223:)
11218:2
11214:C
11210:+
11207:x
11202:1
11198:C
11194:(
11191:=
11186:c
11182:y
11168:c
11164:b
11144:)
11138:c
11135:4
11127:2
11123:b
11114:b
11110:(
11103:2
11100:x
11091:e
11085:2
11081:C
11077:+
11071:)
11065:c
11062:4
11054:2
11050:b
11044:+
11041:b
11037:(
11030:2
11027:x
11018:e
11012:1
11008:C
11004:=
10999:c
10995:y
10981:c
10977:b
10959:p
10955:y
10951:+
10946:c
10942:y
10938:=
10935:y
10922:x
10920:(
10918:r
10909:p
10907:y
10889:x
10884:j
10875:e
10860:,
10858:e
10853:c
10851:y
10845:c
10843:y
10826:)
10823:x
10820:(
10817:r
10814:=
10811:y
10808:c
10805:+
10799:x
10796:d
10791:y
10788:d
10782:b
10779:+
10771:2
10767:x
10763:d
10758:y
10753:2
10749:d
10719:]
10713:2
10709:C
10705:+
10702:x
10697:1
10693:C
10689:+
10683:d
10679:)
10672:d
10668:)
10662:(
10659:Q
10651:d
10647:)
10641:(
10638:P
10624:e
10609:(
10603:x
10594:[
10585:d
10581:)
10575:(
10572:P
10567:x
10555:e
10551:=
10548:y
10522:d
10518:)
10512:(
10509:P
10504:x
10495:e
10471:)
10468:x
10465:(
10462:q
10459:=
10456:y
10452:)
10448:)
10445:x
10442:(
10435:p
10431:+
10426:2
10422:)
10418:x
10415:(
10412:p
10408:(
10404:+
10398:x
10395:d
10390:y
10387:d
10381:)
10378:x
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10229:[
10220:d
10216:)
10210:(
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10202:x
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10186:=
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10160:.
10152:d
10148:)
10142:(
10139:P
10134:x
10125:e
10101:)
10098:x
10095:(
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10089:=
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10080:x
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9936:d
9928:y
9917:=
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9888:2
9883:)
9877:x
9874:d
9869:y
9866:d
9860:(
9852:x
9849:d
9845:d
9840:=
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9828:x
9824:d
9819:y
9814:2
9810:d
9800:x
9797:d
9792:y
9789:d
9783:2
9769:/
9765:2
9748:)
9745:y
9742:(
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9736:=
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9724:x
9720:d
9715:y
9710:2
9706:d
9653:d
9649:)
9643:,
9640:x
9637:(
9634:N
9631:)
9625:,
9622:x
9619:(
9611:y
9600:x
9585:)
9582:y
9579:,
9576:x
9573:(
9570:M
9567:=
9564:)
9561:x
9558:(
9555:X
9532:d
9528:)
9525:y
9522:,
9516:(
9513:M
9510:)
9507:y
9504:,
9498:(
9490:x
9479:y
9464:)
9461:y
9458:,
9455:x
9452:(
9449:N
9446:=
9443:)
9440:y
9437:(
9434:Y
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9405:=
9399:d
9395:)
9389:(
9386:X
9381:x
9373:+
9367:d
9363:)
9357:,
9354:x
9351:(
9348:N
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9339:,
9336:x
9333:(
9325:y
9313:=
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9299:)
9293:(
9290:Y
9285:y
9277:+
9271:d
9267:)
9264:y
9261:,
9255:(
9252:M
9249:)
9246:y
9243:,
9237:(
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9217:=
9214:)
9211:y
9208:,
9205:x
9202:(
9199:F
9184:)
9182:y
9178:x
9176:(
9174:μ
9153:x
9145:)
9142:N
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9113:)
9110:M
9104:(
9087:)
9085:y
9081:x
9079:(
9077:μ
9054:x
9046:N
9031:y
9023:M
8991:0
8988:=
8981:x
8978:d
8974:)
8971:y
8968:,
8965:x
8962:(
8959:N
8956:+
8953:y
8950:d
8946:)
8943:y
8940:,
8937:x
8934:(
8931:M
8924:0
8921:=
8914:)
8911:y
8908:,
8905:x
8902:(
8899:N
8896:+
8890:x
8887:d
8882:y
8879:d
8873:)
8870:y
8867:,
8864:x
8861:(
8858:M
8822:d
8818:)
8812:,
8809:x
8806:(
8803:N
8798:y
8787:x
8772:)
8769:y
8766:,
8763:x
8760:(
8757:M
8754:=
8751:)
8748:x
8745:(
8742:X
8719:d
8715:)
8712:y
8709:,
8703:(
8700:M
8695:x
8684:y
8669:)
8666:y
8663:,
8660:x
8657:(
8654:N
8651:=
8648:)
8645:y
8642:(
8639:Y
8614:C
8611:=
8605:d
8601:)
8595:(
8592:X
8587:x
8579:+
8573:d
8569:)
8563:,
8560:x
8557:(
8554:N
8549:y
8541:=
8528:d
8524:)
8518:(
8515:Y
8510:y
8502:+
8496:d
8492:)
8489:y
8486:,
8480:(
8477:M
8472:x
8464:=
8457:)
8454:y
8451:,
8448:x
8445:(
8442:F
8410:x
8402:N
8393:=
8387:y
8379:M
8347:0
8344:=
8337:x
8334:d
8330:)
8327:y
8324:,
8321:x
8318:(
8315:N
8312:+
8309:y
8306:d
8302:)
8299:y
8296:,
8293:x
8290:(
8287:M
8280:0
8277:=
8270:)
8267:y
8264:,
8261:x
8258:(
8255:N
8252:+
8246:x
8243:d
8238:y
8235:d
8229:)
8226:y
8223:,
8220:x
8217:(
8214:M
8190:C
8182:M
8178:N
8159:]
8156:)
8150:(
8147:M
8141:)
8135:(
8132:N
8129:[
8118:d
8114:)
8108:(
8105:N
8097:y
8094:x
8086:=
8083:)
8080:x
8077:C
8074:(
8034:0
8031:=
8024:y
8021:d
8017:)
8014:y
8011:x
8008:(
8005:N
8002:x
7999:+
7996:x
7993:d
7989:)
7986:y
7983:x
7980:(
7977:M
7974:y
7967:0
7964:=
7954:x
7951:d
7946:y
7943:d
7936:)
7933:y
7930:x
7927:(
7924:N
7921:x
7918:+
7915:)
7912:y
7909:x
7906:(
7903:M
7900:y
7862:)
7856:(
7853:F
7845:d
7837:x
7833:/
7829:y
7821:=
7818:)
7815:x
7812:C
7809:(
7791:x
7787:u
7765:)
7760:x
7757:y
7752:(
7748:F
7745:=
7739:x
7736:d
7731:y
7728:d
7680:C
7677:=
7671:d
7667:)
7661:(
7658:Q
7653:x
7645:+
7639:d
7635:)
7629:(
7626:P
7621:y
7587:0
7584:=
7577:x
7574:d
7570:)
7567:x
7564:(
7561:Q
7558:+
7555:y
7552:d
7548:)
7545:y
7542:(
7539:P
7532:0
7529:=
7522:)
7519:x
7516:(
7513:Q
7510:+
7504:x
7501:d
7496:y
7493:d
7487:)
7484:y
7481:(
7478:P
7464:y
7460:x
7442:C
7439:+
7433:)
7427:(
7424:F
7416:d
7408:y
7400:=
7397:x
7385:F
7361:x
7358:d
7354:)
7351:y
7348:(
7345:F
7342:=
7335:y
7332:d
7325:)
7322:y
7319:(
7316:F
7313:=
7303:x
7300:d
7295:y
7292:d
7275:y
7257:C
7254:+
7248:d
7244:)
7238:(
7235:F
7230:x
7222:=
7219:y
7190:x
7187:d
7183:)
7180:x
7177:(
7174:F
7171:=
7164:y
7161:d
7154:)
7151:x
7148:(
7145:F
7142:=
7132:x
7129:d
7124:y
7121:d
7104:x
7086:C
7083:=
7077:d
7070:)
7064:(
7059:1
7055:Q
7049:)
7043:(
7038:2
7034:Q
7025:y
7017:+
7011:d
7004:)
6998:(
6993:2
6989:P
6983:)
6977:(
6972:1
6968:P
6959:x
6942:1
6939:Q
6936:2
6933:P
6912:0
6909:=
6902:y
6899:d
6895:)
6892:y
6889:(
6884:2
6880:Q
6876:)
6873:x
6870:(
6865:2
6861:P
6857:+
6854:x
6851:d
6847:)
6844:y
6841:(
6836:1
6832:Q
6828:)
6825:x
6822:(
6817:1
6813:P
6805:0
6802:=
6792:x
6789:d
6784:y
6781:d
6774:)
6771:y
6768:(
6763:2
6759:Q
6755:)
6752:x
6749:(
6744:2
6740:P
6736:+
6733:)
6730:y
6727:(
6722:1
6718:Q
6714:)
6711:x
6708:(
6703:1
6699:P
6683:y
6679:x
6652:x
6648:λ
6639:λ
6635:)
6633:λ
6631:(
6629:F
6619:λ
6617:(
6615:F
6596:2
6593:C
6589:1
6586:C
6580:c
6574:b
6568:y
6562:x
6553:)
6551:y
6549:,
6547:x
6545:(
6543:N
6538:)
6536:y
6534:,
6532:x
6530:(
6528:M
6523:)
6521:y
6519:(
6517:Q
6512:)
6510:y
6508:(
6506:P
6501:)
6499:x
6497:(
6495:Q
6490:)
6488:x
6486:(
6484:P
6458:.
6455:)
6452:)
6447:n
6443:y
6439:,
6433:,
6428:1
6424:y
6420:,
6417:x
6414:(
6411:F
6408:,
6403:n
6399:y
6395:,
6389:,
6384:2
6380:y
6376:(
6373:=
6370:)
6365:n
6361:y
6357:,
6351:,
6346:1
6342:y
6338:,
6335:x
6332:(
6328:F
6323:,
6320:)
6315:n
6311:y
6307:,
6301:,
6296:1
6292:y
6288:(
6285:=
6281:y
6254:)
6250:y
6246:,
6243:x
6240:(
6236:F
6232:=
6224:y
6192:.
6189:)
6184:n
6180:y
6176:,
6170:,
6165:1
6161:y
6157:,
6154:x
6151:(
6148:F
6143:=
6133:n
6129:y
6119:n
6115:y
6109:=
6099:1
6093:n
6089:y
6070:3
6066:y
6060:=
6050:2
6046:y
6036:2
6032:y
6026:=
6016:1
6012:y
5994:n
5990:n
5986:i
5968:.
5963:)
5960:1
5954:i
5951:(
5947:y
5943:=
5938:i
5934:y
5920:n
5901:)
5898:n
5895:(
5891:y
5887:=
5883:)
5877:)
5874:1
5868:n
5865:(
5861:y
5854:,
5845:,
5838:y
5834:,
5827:y
5823:,
5820:y
5817:,
5814:x
5810:(
5806:F
5793:n
5756:,
5744:)
5741:x
5738:(
5735:y
5721:x
5714:x
5685:R
5660:,
5657:)
5652:0
5648:y
5643:/
5639:1
5636:+
5631:0
5627:x
5623:(
5616:R
5605:y
5581:0
5573:0
5569:y
5562:)
5555:+
5552:,
5545:0
5541:y
5537:1
5532:+
5527:0
5523:x
5518:(
5510:0
5502:0
5498:y
5491:)
5483:0
5479:y
5475:1
5470:+
5465:0
5461:x
5457:,
5447:(
5439:0
5436:=
5431:0
5427:y
5420:R
5413:{
5382:1
5379:+
5374:0
5370:y
5366:)
5363:x
5355:0
5351:x
5347:(
5341:0
5337:y
5331:=
5328:)
5325:x
5322:(
5319:y
5295:R
5281:C
5277:y
5271:(
5269:F
5250:2
5246:y
5242:=
5235:y
5215:0
5212:y
5208:0
5205:x
5188:R
5135:F
5097:)
5094:x
5091:(
5088:y
5077:x
5070:x
5034:)
5031:x
5028:(
5025:y
5011:x
5004:x
4960:x
4930:I
4896:I
4887:0
4883:x
4879:,
4876:}
4867:{
4860:R
4847:x
4843:,
4840:)
4835:+
4831:x
4827:,
4818:x
4814:(
4811:=
4802:I
4788:0
4785:y
4781:0
4778:x
4763:y
4759:x
4757:(
4755:F
4751:F
4742:R
4738:h
4723:]
4720:a
4717:+
4712:0
4708:x
4704:,
4701:a
4693:0
4689:x
4685:[
4679:]
4676:h
4673:+
4668:0
4664:x
4660:,
4657:h
4649:0
4645:x
4641:[
4638:=
4635:I
4617:×
4612:R
4608:b
4604:a
4595:b
4591:a
4573:]
4570:b
4567:+
4562:0
4558:y
4554:,
4551:b
4543:0
4539:y
4535:[
4529:]
4526:a
4523:+
4518:0
4514:x
4510:,
4507:a
4499:0
4495:x
4491:[
4488:=
4485:R
4475:y
4471:F
4467:F
4453:)
4448:0
4444:x
4440:(
4437:y
4434:=
4429:0
4425:y
4420:,
4416:)
4413:y
4410:,
4407:x
4404:(
4401:F
4398:=
4391:y
4199:f
4152:n
4090:2
4085:)
4080:|
4074:2
4071:x
4063:1
4059:|
4055:+
4050:2
4047:x
4039:1
4035:(
4028:4
4025:1
4020:=
4017:)
4014:x
4011:(
4008:y
3982:1
3979:=
3976:)
3973:0
3970:(
3967:y
3962:,
3956:|
3952:y
3948:|
3942:)
3939:y
3936:(
3925:=
3918:y
3849:n
3845:n
3830:R
3808:.
3805:I
3799:x
3795:)
3792:x
3789:(
3786:v
3783:=
3780:)
3777:x
3774:(
3771:u
3757:J
3753:I
3748:v
3740:u
3735:R
3731:R
3727:I
3723:v
3717:R
3713:R
3709:J
3705:u
3687:.
3684:I
3678:x
3674:0
3671:=
3668:)
3663:)
3660:n
3657:(
3653:u
3646:,
3637:,
3630:u
3626:,
3623:u
3620:,
3617:x
3614:(
3611:F
3598:I
3594:n
3590:u
3586:F
3574:I
3569:R
3565:R
3561:I
3557:u
3539:0
3536:=
3532:)
3526:)
3523:n
3520:(
3516:y
3512:,
3506:,
3499:y
3495:,
3492:y
3489:,
3486:x
3482:(
3478:F
3419:v
3410:)
3406:v
3402:,
3398:u
3394:,
3391:x
3388:(
3384:F
3353:0
3349:=
3345:)
3336:y
3331:,
3327:y
3323:,
3320:x
3316:(
3311:F
3285:)
3279:0
3265:0
3258:0
3252:(
3247:=
3242:)
3236:)
3231:)
3228:n
3225:(
3220:y
3215:,
3209:,
3201:y
3196:,
3188:y
3183:,
3179:y
3175:,
3172:x
3169:(
3164:m
3160:f
3145:)
3140:)
3137:n
3134:(
3129:y
3124:,
3118:,
3110:y
3105:,
3097:y
3092:,
3088:y
3084:,
3081:x
3078:(
3073:2
3069:f
3061:)
3056:)
3053:n
3050:(
3045:y
3040:,
3034:,
3026:y
3021:,
3013:y
3008:,
3004:y
3000:,
2997:x
2994:(
2989:1
2985:f
2978:(
2959:0
2941:0
2937:=
2933:)
2927:)
2924:n
2921:(
2916:y
2911:,
2905:,
2897:y
2892:,
2884:y
2879:,
2875:y
2871:,
2868:x
2864:(
2859:F
2826:)
2819:)
2813:)
2810:1
2804:n
2801:(
2796:y
2791:,
2785:,
2777:y
2772:,
2764:y
2759:,
2755:y
2751:,
2748:x
2744:(
2738:m
2734:f
2718:)
2712:)
2709:1
2703:n
2700:(
2695:y
2690:,
2684:,
2676:y
2671:,
2663:y
2658:,
2654:y
2650:,
2647:x
2643:(
2637:2
2633:f
2624:)
2618:)
2615:1
2609:n
2606:(
2601:y
2596:,
2590:,
2582:y
2577:,
2569:y
2564:,
2560:y
2556:,
2553:x
2549:(
2543:1
2539:f
2532:(
2527:=
2522:)
2514:)
2511:n
2508:(
2503:m
2499:y
2482:)
2479:n
2476:(
2471:2
2467:y
2457:)
2454:n
2451:(
2446:1
2442:y
2435:(
2416:m
2409:n
2384:)
2378:)
2375:1
2369:n
2366:(
2361:y
2356:,
2350:,
2342:y
2337:,
2329:y
2324:,
2320:y
2316:,
2313:x
2309:(
2304:F
2300:=
2295:)
2292:n
2289:(
2284:y
2269:y
2261:F
2257:x
2255:(
2253:y
2249:y
2217:x
2215:(
2213:r
2196:.
2192:y
2181:x
2179:(
2177:r
2153:x
2151:(
2149:r
2145:x
2141:)
2139:x
2135:r
2130:)
2128:x
2126:(
2120:i
2115:a
2098:)
2095:x
2092:(
2089:r
2086:+
2081:)
2078:i
2075:(
2071:y
2067:)
2064:x
2061:(
2056:i
2052:a
2046:1
2040:n
2035:0
2032:=
2029:i
2021:=
2016:)
2013:n
2010:(
2006:y
1992:y
1973:F
1951:.
1948:x
1915:0
1912:=
1908:)
1902:)
1899:n
1896:(
1892:y
1885:,
1876:,
1869:y
1865:,
1858:y
1854:,
1851:y
1848:,
1845:x
1841:(
1837:F
1824:n
1791:)
1788:n
1785:(
1781:y
1777:=
1773:)
1767:)
1764:1
1758:n
1755:(
1751:y
1747:,
1741:,
1734:y
1730:,
1727:y
1724:,
1721:x
1717:(
1713:F
1700:y
1696:y
1692:x
1688:F
1660:)
1654:.
1651:.
1648:.
1644:y
1639:,
1630:y
1624:,
1615:y
1609:(
1592:)
1590:y
1586:y
1582:y
1580:(
1569:)
1559:/
1555:y
1552:d
1536:/
1532:y
1529:d
1513:/
1503:(
1492:x
1485:)
1483:x
1481:(
1479:f
1475:y
1466:y
1454:t
1452:(
1450:x
1448:(
1446:F
1442:t
1440:(
1438:x
1434:t
1430:t
1428:(
1426:x
1422:F
1418:m
1396:)
1393:)
1390:t
1387:(
1384:x
1381:(
1378:F
1375:=
1367:2
1363:t
1358:d
1352:)
1349:t
1346:(
1343:x
1338:2
1333:d
1325:m
1312:F
1308:t
1304:x
1153:x
1149:y
1131:)
1128:n
1125:(
1121:y
1117:,
1111:,
1104:y
1075:)
1072:x
1069:(
1066:b
1042:)
1039:x
1036:(
1031:n
1027:a
1002:)
999:x
996:(
991:0
987:a
961:,
958:0
955:=
952:)
949:x
946:(
943:b
940:+
935:)
932:n
929:(
925:y
921:)
918:x
915:(
910:n
906:a
902:+
896:+
889:y
885:)
882:x
879:(
874:2
870:a
866:+
859:y
855:)
852:x
849:(
844:1
840:a
836:+
833:y
830:)
827:x
824:(
819:0
815:a
749:(
730:e
723:t
716:v
529:)
525:(
20:)
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