5520:
7881:
5144:
5891:
7582:
2745:
8646:
8697:
Similarly to the algebraic case, the theory allows deciding which equations may be solved by quadrature, and if possible solving them. However, for both theories, the necessary computations are extremely difficult, even with the most powerful computers.
1348:
7335:
A system of linear differential equations consists of several linear differential equations that involve several unknown functions. In general one restricts the study to systems such that the number of unknown functions equals the number of equations.
1147:. In the ordinary case, this vector space has a finite dimension, equal to the order of the equation. All solutions of a linear differential equation are found by adding to a particular solution any solution of the associated homogeneous equation.
3614:
5515:{\displaystyle {\begin{aligned}0&=u'_{1}y_{1}+u'_{2}y_{2}+\cdots +u'_{n}y_{n}\\0&=u'_{1}y'_{1}+u'_{2}y'_{2}+\cdots +u'_{n}y'_{n}\\&\;\;\vdots \\0&=u'_{1}y_{1}^{(n-2)}+u'_{2}y_{2}^{(n-2)}+\cdots +u'_{n}y_{n}^{(n-2)},\end{aligned}}}
3672:, the sum of the multiplicities of the roots of a polynomial equals the degree of the polynomial, the number of above solutions equals the order of the differential equation, and these solutions form a base of the vector space of the solutions.
6140:
As antiderivatives are defined up to the addition of a constant, one finds again that the general solution of the non-homogeneous equation is the sum of an arbitrary solution and the general solution of the associated homogeneous equation.
1922:
8384:
1764:
890:
7339:
An arbitrary linear ordinary differential equation and a system of such equations can be converted into a first order system of linear differential equations by adding variables for all but the highest order derivatives. That is, if
3084:
1504:
4759:
8872:
5651:
2561:
2375:
6030:
4386:
992:. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two,
5004:
2597:
4488:
7587:
5635:
8010:
6595:
2957:
2167:
6681:
8481:
2853:
9238:
6316:
8476:
7401:
6963:
1241:
5149:
4171:
7876:{\displaystyle {\begin{aligned}y_{1}'(x)&=b_{1}(x)+a_{1,1}(x)y_{1}+\cdots +a_{1,n}(x)y_{n}\\&\;\;\vdots \\y_{n}'(x)&=b_{n}(x)+a_{n,1}(x)y_{1}+\cdots +a_{n,n}(x)y_{n},\end{aligned}}}
7314:
3263:
8056:
3336:
8247:
6848:
5079:
6513:
3500:
6912:
6758:
3992:
3164:
1604:
As the sum of two linear operators is a linear operator, as well as the product (on the left) of a linear operator by a differentiable function, the linear differential operators form a
6237:
4264:
6362:
1232:
8926:
8162:
7549:
3662:
3485:
7451:
3927:
3869:
8267:
7107:
7054:
3393:
1630:
6999:
8413:
3811:
3756:
1376:
7210:
9075:, fast computation of Taylor series (thanks of the recurrence relation on its coefficients), evaluation to a high precision with certified bound of the approximation error,
8014:
The solving method is similar to that of a single first order linear differential equations, but with complications stemming from noncommutativity of matrix multiplication.
7493:
7248:
7151:
4037:
6409:
4593:
1963:
7947:
8719:
7910:
8099:
1769:
1116:
in the unknown function and its derivatives. The equation obtained by replacing, in a linear differential equation, the constant term by the zero function is the
740:
9359:
4292:
4395:
984:
A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by
5920:
5533:
4886:
1051:. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of
9754:
19:
This article is about linear differential equations with one independent variable. For similar equations with two or more independent variables, see
2453:
2267:
9759:
8955:
Most functions that are commonly considered in mathematics are holonomic or quotients of holonomic functions. In fact, holonomic functions include
3675:
In the common case where the coefficients of the equation are real, it is generally more convenient to have a basis of the solutions consisting of
2861:
2063:
704:
6606:
9622:
9096:
9052:
is zero). There are efficient algorithms for both conversions, that is for computing the recurrence relation from the differential equation, and
445:
2763:
9749:
8690:
of degree at least five cannot, in general, be solved by radicals. This analogy extends to the proof methods and motivates the denomination of
7966:
2395:
are (real or complex) numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.
20:
9059:
It follows that, if one represents (in a computer) holonomic functions by their defining differential equations and initial conditions, most
1373:
of basic differential operators, with differentiable functions as coefficients. In the univariate case, a linear operator has thus the form
8425:
6255:
9352:
4107:
1143:
of a differential equation is a function that satisfies the equation. The solutions of a homogeneous linear differential equation form a
7253:
3184:
2969:
distinct solutions that are not necessarily real, even if the coefficients of the equation are real. These solutions can be shown to be
9409:
8716:
are examples of equations of any order, with variable coefficients, that can be solved explicitly. These are the equations of the form
7576:
3268:
188:
8181:
6917:
5886:{\displaystyle y^{(n)}=u_{1}y_{1}^{(n)}+\cdots +u_{n}y_{n}^{(n)}+u'_{1}y_{1}^{(n-1)}+u'_{2}y_{2}^{(n-1)}+\cdots +u'_{n}y_{n}^{(n-1)}.}
5009:
9522:
9026:
for computing the differential equation of the result of any of these operations, knowing the differential equations of the input.
6686:
6520:
319:
8020:
6783:
3089:
6436:
9345:
2740:{\displaystyle a_{0}e^{\alpha x}+a_{1}\alpha e^{\alpha x}+a_{2}\alpha ^{2}e^{\alpha x}+\cdots +a_{n}\alpha ^{n}e^{\alpha x}=0.}
9566:
4193:
3010:
2030:
1967:
There may be several variants to this notation; in particular the variable of differentiation may appear explicitly or not in
1593:, if one needs to specify the variable (this must not be confused with a multiplication). A linear differential operator is a
360:
9460:
9293:
5119:
are arbitrary constants. The method of variation of constants takes its name from the following idea. Instead of considering
1191:
250:
7575:
differential equations may normally be solved for the derivatives of the unknown functions. If it is not the case this is a
9744:
697:
268:
152:
9654:
6163:
9669:
9450:
4858:
571:
225:
3355:
6968:
2856:
9455:
9419:
9311:
9275:
7156:
1108:
246:
198:
9022:
of holonomic functions are holonomic. Moreover, these closure properties are effective, in the sense that there are
8641:{\displaystyle \mathbf {y} (x)=U(x)U^{-1}(x_{0})\mathbf {y_{0}} +U(x)\int _{x_{0}}^{x}U^{-1}(t)\mathbf {b} (t)\,dt.}
9517:
9429:
9044:
at a point of a holonomic function form a holonomic sequence. Conversely, if the sequence of the coefficients of a
7330:
7215:
7059:
7006:
3940:
314:
233:
208:
9866:
9485:
8988:
7111:
3997:
1036:
972:(PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are
690:
9800:
9399:
8952:, is a function that is a solution of a homogeneous linear differential equation with polynomial coefficients.
8249:
In the general case there is no closed-form solution for the homogeneous equation, and one has to use either a
6853:
6375:
3669:
961:
625:
178:
9556:
6149:
The general form of a linear ordinary differential equation of order 1, after dividing out the coefficient of
4802:
There are several methods for solving such an equation. The best method depends on the nature of the function
1927:
1354:
variables. The basic differential operators include the derivative of order 0, which is the identity mapping.
1343:{\displaystyle {\frac {\partial ^{i_{1}+\cdots +i_{n}}}{\partial x_{1}^{i_{1}}\cdots \partial x_{n}^{i_{n}}}}}
350:
9571:
9414:
9404:
6329:
969:
365:
193:
183:
9162:, and try to modify the left side so it becomes a derivative. Specifically, we seek an "integrating factor"
7345:
9394:
8992:
8879:
8130:
7324:
4811:
640:
491:
394:
281:
9659:
9529:
9490:
7410:
3874:
3816:
3629:
3452:
326:
241:
3609:{\displaystyle \left({\frac {d}{dx}}-\alpha \right)\left(x^{k}e^{\alpha x}\right)=kx^{k-1}e^{\alpha x},}
2187:
are arbitrary numbers. Typically, the hypotheses of Carathéodory's theorem are satisfied in an interval
9111:
8691:
8676:
531:
399:
9495:
9251:
Benoit, A., Chyzak, F., Darrasse, A., Gerhold, S., Mezzarobba, M., & Salvy, B. (2010, September).
8389:
3761:
3706:
9780:
9699:
9080:
8713:
8664:
7502:
6424:
For the general non-homogeneous equation, it is useful to multiply both sides of the equation by the
4571:
502:
480:
9835:
9805:
645:
9694:
8683:
8656:
985:
496:
404:
9704:
9674:
9664:
9561:
9116:
9029:
Usefulness of the concept of holonomic functions results of
Zeilberger's theorem, which follows.
9004:
8416:
6323:
2974:
2008:
1170:
1048:
936:
576:
566:
558:
514:
355:
7579:, and this is a different theory. Therefore, the systems that are considered here have the form
5137:
as constants, they can be considered as unknown functions that have to be determined for making
9257:. In International Congress on Mathematical Software (pp. 35-41). Springer, Berlin, Heidelberg.
8702:
6425:
5527:
4877:
2989:
1113:
993:
389:
16:
Differential equations that are linear with respect to the unknown function and its derivatives
7460:
2996:
of solutions of the differential equation (that is, the kernel of the differential operator).
996:
allows deciding whether there are solutions in terms of integrals, and computing them if any.
9820:
9815:
9709:
9633:
9612:
9480:
9475:
9470:
9465:
9389:
9368:
9145:
9049:
8701:
Nevertheless, the case of order two with rational coefficients has been completely solved by
7919:
3444:. Thus, applying the differential operator of the equation is equivalent with applying first
1156:
1094:, which does not depend on the unknown function and its derivatives, is sometimes called the
726:
635:
620:
509:
452:
434:
273:
29:
4552:, respectively. This results in a linear system of two linear equations in the two unknowns
9714:
9638:
9607:
8968:
8379:{\displaystyle \mathbf {y} (x)=U(x)\mathbf {y_{0}} +U(x)\int U^{-1}(x)\mathbf {b} (x)\,dx,}
7888:
3676:
2970:
2403:
1759:{\displaystyle L=a_{0}(x)+a_{1}(x){\frac {d}{dx}}+\cdots +a_{n}(x){\frac {d^{n}}{dx^{n}}},}
1598:
1064:
1020:
521:
457:
430:
8075:
8058:
be the homogeneous equation associated to the above matrix equation. Its solutions form a
8:
9724:
9617:
9511:
9084:
9076:
9068:
9037:
3352:, more linearly independent solutions are needed for having a basis. These have the form
1499:{\displaystyle a_{0}(x)+a_{1}(x){\frac {d}{dx}}+\cdots +a_{n}(x){\frac {d^{n}}{dx^{n}}},}
1076:
1060:
553:
538:
439:
303:
142:
109:
100:
3348:, the preceding provides a complete basis of the solutions vector space. In the case of
9845:
9684:
9679:
9602:
9434:
8960:
8945:
8939:
8687:
8171:
5141:
a solution of the non-homogeneous equation. For this purpose, one adds the constraints
4870:
4862:
4389:
3700:
1621:
1370:
1181:
1099:
1004:
973:
548:
543:
426:
9307:
9289:
9271:
9253:
9106:
9101:
9072:
8984:
4276:
1174:
1129:
730:
605:
384:
119:
8655:
A linear ordinary equation of order one with variable coefficients may be solved by
1627:
The language of operators allows a compact writing for differentiable equations: if
660:
9830:
9810:
9011:
8996:
8254:
8250:
2447:, and this allows solving homogeneous linear differential equations rather easily.
2398:
The study of these differential equations with constant coefficients dates back to
1016:
670:
655:
2563:
be a homogeneous linear differential equation with constant coefficients (that is
9790:
9719:
9000:
8980:
4754:{\displaystyle y^{(n)}(x)+a_{1}y^{(n-1)}(x)+\cdots +a_{n-1}y'(x)+a_{n}y(x)=f(x),}
3699:
is also a root, of the same multiplicity. Thus a real basis is obtained by using
1616:(depending on the nature of the functions that are considered). They form also a
1594:
1044:
610:
526:
53:
9795:
9326:
8668:
665:
9785:
9592:
9019:
8867:{\displaystyle x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\cdots +a_{0}y(x)=0,}
8672:
8660:
8123:
6365:
6080:
6076:
3937:
A homogeneous linear differential equation of the second order may be written
2962:
2399:
1613:
1102:), even when this term is a non-constant function. If the constant term is the
1056:
1040:
1012:
989:
630:
615:
421:
409:
128:
4067:. In all three cases, the general solution depends on two arbitrary constants
3679:. Such a basis may be obtained from the preceding basis by remarking that, if
9860:
9041:
8067:
3494:
3349:
1917:{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}=b(x)}
1558:
1103:
6772:(changing of antiderivative amounts to change the constant of integration).
885:{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)}
9840:
9775:
9689:
9177:
such that multiplying by it makes the left side equal to the derivative of
9045:
8059:
6600:
5523:
2993:
2012:
1605:
1144:
1067:, and numerical evaluation to any precision, with a certified error bound.
650:
600:
486:
114:
8663:. This is not the case for order at least two. This is the main result of
4810:
is a linear combination of exponential and sinusoidal functions, then the
9597:
9337:
8104:
4053:, there are three cases for the solutions, depending on the discriminant
4050:
3417:. For proving that these functions are solutions, one may remark that if
3345:
1617:
1609:
718:
58:
9242:. Journal of computational and applied mathematics. 32.3 (1990): 321-368
9048:
is holonomic, then the series defines a holonomic function (even if the
4381:{\displaystyle c_{1}e^{(\alpha +\beta i)x}+c_{2}e^{(\alpha -\beta i)x},}
9825:
9064:
9015:
8956:
1235:
1008:
1000:
675:
4581:
9587:
9023:
8964:
7405:
appear in an equation, one may replace them by new unknown functions
6075:, and their derivatives). This system can be solved by any method of
1024:
416:
137:
80:
70:
21:
Partial differential equation § Linear equations of second order
8682:
The impossibility of solving by quadrature can be compared with the
4483:{\displaystyle e^{\alpha x}(c_{1}\cos(\beta x)+c_{2}\sin(\beta x)).}
9627:
9060:
4869:
satisfies a homogeneous linear differential equation, typically, a
1052:
734:
6144:
6025:{\displaystyle f=u'_{1}y_{1}^{(n-1)}+\cdots +u'_{n}y_{n}^{(n-1)}.}
5899:
and its derivatives by these expressions, and using the fact that
2255:
9331:
9265:
9063:
operations can be done automatically on these functions, such as
5630:{\displaystyle y^{(i)}=u_{1}y_{1}^{(i)}+\cdots +u_{n}y_{n}^{(i)}}
4999:{\displaystyle y^{(n)}+a_{1}y^{(n-1)}+\cdots +a_{n-1}y'+a_{n}y=0}
2855:
of the differential equation, which is the left-hand side of the
91:
86:
75:
999:
The solutions of homogeneous linear differential equations with
8976:
7003:
Dividing the original equation by one of these solutions gives
4857:
a constant (which need not be the same in each term), then the
1132:
appear as coefficients in the associated homogeneous equation.
1032:
9334:. Automatic and interactive study of many holonomic functions.
9087:
at infinity and near singularities, proof of identities, etc.
8659:, which means that the solutions may be expressed in terms of
5917:
are solutions of the original homogeneous equation, one gets
4578:
for the solution of the DEQ and its derivative are specified.
2556:{\displaystyle a_{0}y+a_{1}y'+a_{2}y''+\cdots +a_{n}y^{(n)}=0}
2370:{\displaystyle a_{0}y+a_{1}y'+a_{2}y''+\cdots +a_{n}y^{(n)}=0}
988:, which means that the solutions may be expressed in terms of
6590:{\displaystyle -fe^{-F}={\tfrac {d}{dx}}\left(e^{-F}\right),}
4883:
The general solution of the associated homogeneous equation
2952:{\displaystyle a_{0}+a_{1}t+a_{2}t^{2}+\cdots +a_{n}t^{n}=0.}
2162:{\displaystyle S_{0}(x)+c_{1}S_{1}(x)+\cdots +c_{n}S_{n}(x),}
9239:
A holonomic systems approach to special functions identities
7955:. In matrix notation, this system may be written (omitting "
6676:{\displaystyle {\frac {d}{dx}}\left(ye^{-F}\right)=ge^{-F}.}
6372:. Thus, the general solution of the homogeneous equation is
4096:, the characteristic polynomial has two distinct real roots
2015:
of the solutions of the (homogeneous) differential equation
8972:
3421:
is a root of the characteristic polynomial of multiplicity
3403:
is a root of the characteristic polynomial of multiplicity
1028:
8650:
8264:, the general solution of the non-homogeneous equation is
7318:
4530:, one equates the values of the above general solution at
2848:{\displaystyle a_{0}+a_{1}t+a_{2}t^{2}+\cdots +a_{n}t^{n}}
2025:
In the case of an ordinary differential operator of order
1533:
are differentiable functions, and the nonnegative integer
1007:. This class of functions is stable under sums, products,
8478:
the solution that satisfies these initial conditions is
8005:{\displaystyle \mathbf {y} '=A\mathbf {y} +\mathbf {b} .}
4570:. Solving this system gives the solution for a so-called
3344:
In the case where the characteristic polynomial has only
2584:
Searching solutions of this equation that have the form
2033:
implies that, under very mild conditions, the kernel of
1079:
that appears in a (linear) differential equation is the
733:
in the unknown function and its derivatives, that is an
9254:
The dynamic dictionary of mathematical functions (DDMF)
6433:
of a solution of the homogeneous equation. This gives
9040:
with polynomial coefficients. The coefficients of the
8134:
6547:
6339:
3632:
3455:
1766:
is a linear differential operator, then the equation
1568:
be a linear differential operator. The application of
1180:, or, in the case of several variables, to one of its
949:
are the successive derivatives of an unknown function
8882:
8722:
8484:
8471:{\displaystyle \mathbf {y} (x_{0})=\mathbf {y} _{0},}
8428:
8392:
8270:
8184:
8133:
8078:
8023:
7969:
7922:
7891:
7585:
7505:
7463:
7413:
7348:
7256:
7218:
7159:
7114:
7062:
7009:
6971:
6920:
6856:
6786:
6689:
6609:
6523:
6439:
6378:
6332:
6258:
6166:
5923:
5654:
5536:
5147:
5012:
4889:
4596:
4398:
4295:
4196:
4110:
4000:
3943:
3877:
3819:
3764:
3709:
3503:
3358:
3271:
3187:
3092:
3013:
2864:
2766:
2600:
2456:
2270:
2066:
1930:
1772:
1633:
1379:
1244:
1194:
743:
9036:
is a sequence of numbers that may be generated by a
6311:{\displaystyle {\frac {y'}{y}}=f,\qquad \log y=k+F,}
5101:
is a basis of the vector space of the solutions and
4582:
Non-homogeneous equation with constant coefficients
4166:{\displaystyle c_{1}e^{\alpha x}+c_{2}e^{\beta x}.}
3425:, the characteristic polynomial may be factored as
9304:An Introduction to Ordinary Differential Equations
9301:
8920:
8866:
8640:
8470:
8407:
8378:
8241:
8156:
8093:
8050:
8004:
7941:
7904:
7875:
7543:
7487:
7445:
7395:
7309:{\displaystyle y(x)=x^{2}+{\frac {\alpha -1}{x}}.}
7308:
7242:
7204:
7145:
7101:
7048:
6993:
6957:
6906:
6842:
6752:
6675:
6589:
6507:
6403:
6356:
6310:
6231:
6024:
5885:
5629:
5514:
5073:
4998:
4791:is the unknown function (for sake of simplicity, "
4753:
4482:
4380:
4258:
4182:, the characteristic polynomial has a double root
4165:
4031:
3986:
3921:
3863:
3805:
3750:
3656:
3608:
3479:
3387:
3330:
3258:{\displaystyle e^{ix},\;e^{-ix},\;e^{x},\;xe^{x}.}
3257:
3158:
3078:
2951:
2847:
2739:
2555:
2369:
2161:
1957:
1916:
1758:
1597:, since it maps sums to sums and the product by a
1498:
1342:
1226:
884:
9755:List of nonlinear ordinary differential equations
9283:
9266:Birkhoff, Garrett & Rota, Gian-Carlo (1978),
8118:is a matrix of constants, or, more generally, if
4818:is a linear combination of functions of the form
3689:is a root of the characteristic polynomial, then
3331:{\displaystyle \cos x,\;\sin x,\;e^{x},\;xe^{x}.}
9858:
9760:List of nonlinear partial differential equations
8242:{\displaystyle {\frac {d}{dx}}\exp(B)=A\exp(B).}
6958:{\displaystyle {\frac {y'}{y}}=-{\frac {1}{x}},}
5074:{\displaystyle y=u_{1}y_{1}+\cdots +u_{n}y_{n},}
1971:and the right-hand and of the equation, such as
6145:First-order equation with variable coefficients
2411:, which is the unique solution of the equation
2260:A homogeneous linear differential equation has
2256:Homogeneous equation with constant coefficients
1150:
1106:, then the differential equation is said to be
9750:List of linear ordinary differential equations
9306:, Cambridge, UK.: Cambridge University Press,
9288:, Cambridge, UK.: Cambridge University Press,
7567:A linear system of the first order, which has
9353:
9327:http://eqworld.ipmnet.ru/en/solutions/ode.htm
6753:{\displaystyle y=ce^{F}+e^{F}\int ge^{-F}dx,}
3181:(multiplicity 2). The solution basis is thus
698:
9232:
9230:
9228:
8051:{\displaystyle \mathbf {u} '=A\mathbf {u} .}
4806:that makes the equation non-homogeneous. If
4388:which may be rewritten in real terms, using
9332:Dynamic Dictionary of Mathematical Function
8675:, and whose recent developments are called
6843:{\displaystyle y'(x)+{\frac {y(x)}{x}}=3x.}
3159:{\displaystyle z^{4}-2z^{3}+2z^{2}-2z+1=0.}
9367:
9360:
9346:
7728:
7727:
6508:{\displaystyle y'e^{-F}-yfe^{-F}=ge^{-F}.}
5355:
5354:
4590:with constant coefficients may be written
3311:
3297:
3284:
3238:
3224:
3204:
705:
691:
9225:
8628:
8366:
6907:{\displaystyle y'(x)+{\frac {y(x)}{x}}=0}
6346:
2590:is equivalent to searching the constants
2041:, and that the solutions of the equation
2007:of a linear differential operator is its
8708:
6062:whose coefficients are known functions (
4275:, the characteristic polynomial has two
4259:{\displaystyle (c_{1}+c_{2}x)e^{-ax/2}.}
4104:. In this case, the general solution is
3079:{\displaystyle y''''-2y'''+2y''-2y'+y=0}
1015:, and contains many usual functions and
9270:, New York: John Wiley and Sons, Inc.,
8651:Higher order with variable coefficients
7319:System of linear differential equations
6357:{\displaystyle F=\textstyle \int f\,dx}
1227:{\displaystyle {\frac {d^{i}}{dx^{i}}}}
9859:
8933:
7396:{\displaystyle y',y'',\ldots ,y^{(k)}}
6034:This equation and the above ones with
2217:, and there is a positive real number
9341:
8921:{\displaystyle a_{0},\ldots ,a_{n-1}}
8253:, or an approximation method such as
8157:{\displaystyle \textstyle B=\int Adx}
8066:, and are therefore the columns of a
6241:If the equation is homogeneous, i.e.
4861:may be used. Still more general, the
4799:" will be omitted in the following).
3994:and its characteristic polynomial is
3493:as characteristic polynomial. By the
9745:List of named differential equations
8178:. In fact, in these cases, one has
6850:The associated homogeneous equation
6232:{\displaystyle y'(x)=f(x)y(x)+g(x).}
4586:A non-homogeneous equation of order
3932:
3657:{\textstyle {\frac {d}{dx}}-\alpha }
3480:{\textstyle {\frac {d}{dx}}-\alpha }
2977:of the values of these solutions at
1070:
979:
153:List of named differential equations
9670:Method of undetermined coefficients
9451:Dependent and independent variables
9286:The Nature of Mathematical Modeling
9148:technique we write the equation as
8422:If initial conditions are given as
7446:{\displaystyle y_{1},\ldots ,y_{k}}
6038:as left-hand side form a system of
5895:Replacing in the original equation
4859:method of undetermined coefficients
3922:{\displaystyle x^{k}e^{ax}\sin(bx)}
3864:{\displaystyle x^{k}e^{ax}\cos(bx)}
1601:to the product by the same scalar.
939:that do not need to be linear, and
226:Dependent and independent variables
13:
6764:is a constant of integration, and
3388:{\displaystyle x^{k}e^{\alpha x},}
2753:(which is never zero), shows that
1361:(abbreviated, in this article, as
1312:
1284:
1248:
14:
9878:
9320:
9014:; in particular, sums, products,
9010:Holonomic functions have several
7250:one gets the particular solution
6994:{\displaystyle y={\frac {c}{x}}.}
6603:allows rewriting the equation as
6252:, one may rewrite and integrate:
4814:may be used. If, more generally,
3265:A real basis of solution is thus
2011:as a linear mapping, that is the
1098:of the equation (by analogy with
9567:Carathéodory's existence theorem
8615:
8549:
8545:
8486:
8455:
8430:
8408:{\displaystyle \mathbf {y_{0}} }
8399:
8395:
8353:
8306:
8302:
8272:
8041:
8026:
7995:
7987:
7972:
7455:that must satisfy the equations
7331:system of differential equations
3806:{\displaystyle x^{k}e^{(a-ib)x}}
3751:{\displaystyle x^{k}e^{(a+ib)x}}
3086:has the characteristic equation
2031:Carathéodory's existence theorem
361:Carathéodory's existence theorem
9268:Ordinary Differential Equations
7544:{\displaystyle y_{i}'=y_{i+1},}
7205:{\displaystyle y(x)=x^{2}+c/x.}
6283:
4876:The most general method is the
3489:and then the operator that has
2037:is a vector space of dimension
1120:associated homogeneous equation
1037:inverse trigonometric functions
9245:
9138:
9129:
8852:
8846:
8821:
8815:
8810:
8798:
8755:
8749:
8744:
8738:
8625:
8619:
8611:
8605:
8567:
8561:
8540:
8527:
8511:
8505:
8496:
8490:
8447:
8434:
8363:
8357:
8349:
8343:
8324:
8318:
8297:
8291:
8282:
8276:
8233:
8227:
8212:
8206:
8088:
8082:
7853:
7847:
7809:
7803:
7781:
7775:
7755:
7749:
7707:
7701:
7663:
7657:
7635:
7629:
7609:
7603:
7388:
7382:
7266:
7260:
7228:
7222:
7169:
7163:
7073:
7063:
6889:
6883:
6871:
6865:
6819:
6813:
6801:
6795:
6683:Thus, the general solution is
6223:
6217:
6208:
6202:
6196:
6190:
6181:
6175:
6014:
6002:
5965:
5953:
5875:
5863:
5826:
5814:
5783:
5771:
5740:
5734:
5700:
5694:
5666:
5660:
5622:
5616:
5582:
5576:
5548:
5542:
5500:
5488:
5451:
5439:
5408:
5396:
4936:
4924:
4901:
4895:
4853:is a nonnegative integer, and
4745:
4739:
4730:
4724:
4705:
4699:
4663:
4657:
4652:
4640:
4619:
4613:
4608:
4602:
4474:
4471:
4462:
4440:
4431:
4412:
4367:
4352:
4326:
4311:
4289:, and the general solution is
4226:
4197:
4190:, and the general solution is
3916:
3907:
3858:
3849:
3795:
3780:
3740:
3725:
3670:fundamental theorem of algebra
2581:are real or complex numbers).
2542:
2536:
2356:
2350:
2153:
2147:
2115:
2109:
2083:
2077:
1949:
1943:
1911:
1905:
1894:
1888:
1880:
1874:
1844:
1838:
1814:
1808:
1789:
1783:
1721:
1715:
1678:
1672:
1656:
1650:
1461:
1455:
1418:
1412:
1396:
1390:
1124:. A differential equation has
962:ordinary differential equation
879:
873:
862:
856:
848:
842:
815:
809:
785:
779:
760:
754:
448: / Integral solutions
1:
9122:
9097:Continuous-repayment mortgage
8107:is not the zero function. If
7577:differential-algebraic system
7243:{\displaystyle y(1)=\alpha ,}
7102:{\displaystyle (xy)'=3x^{2},}
7049:{\displaystyle xy'+y=3x^{2}.}
4779:are real or complex numbers,
3987:{\displaystyle y''+ay'+by=0,}
3618:and thus one gets zero after
1624:of differentiable functions.
970:partial differential equation
9395:Notation for differentiation
8993:inverse hyperbolic functions
7325:Matrix differential equation
4812:exponential response formula
4534:and its derivative there to
1359:linear differential operator
1350:in the case of functions of
1151:Linear differential operator
966:linear differential equation
723:linear differential equation
492:Exponential response formula
238:Coupled / Decoupled
7:
9491:Exact differential equation
9302:Robinson, James C. (2004),
9090:
8930:are constant coefficients.
7146:{\displaystyle xy=x^{3}+c,}
4880:, which is presented here.
4032:{\displaystyle r^{2}+ar+b.}
1169:is a mapping that maps any
1163:basic differential operator
10:
9883:
9284:Gershenfeld, Neil (1999),
9144:Motivation: In analogy to
9112:Linear difference equation
8937:
8692:differential Galois theory
8677:differential Galois theory
7328:
7322:
7212:For the initial condition
6775:
6421:is an arbitrary constant.
3399:is a nonnegative integer,
1188:. It is commonly denoted
1154:
1083:of the equation. The term
18:
9801:Józef Maria Hoene-Wroński
9781:Gottfried Wilhelm Leibniz
9768:
9737:
9647:
9580:
9572:Cauchy–Kowalevski theorem
9549:
9542:
9504:
9443:
9382:
9375:
6768:is any antiderivative of
6404:{\displaystyle y=ce^{F},}
4574:, in which the values at
3495:exponential shift theorem
2961:When these roots are all
2759:characteristic polynomial
1055:, such as computation of
626:Józef Maria Hoene-Wroński
572:Undetermined coefficients
481:Method of characteristics
366:Cauchy–Kowalevski theorem
9695:Finite difference method
9005:hypergeometric functions
8386:where the column matrix
7488:{\displaystyle y'=y_{1}}
1958:{\displaystyle Ly=b(x).}
1049:hypergeometric functions
1003:coefficients are called
937:differentiable functions
351:Picard–Lindelöf theorem
345:Existence and uniqueness
9675:Variation of parameters
9665:Separation of variables
9562:Peano existence theorem
9557:Picard–Lindelöf theorem
9444:Attributes of variables
9117:Variation of parameters
8686:, which states that an
8667:which was initiated by
8417:constant of integration
7942:{\displaystyle a_{i,j}}
6324:constant of integration
4783:is a given function of
2988:. Together they form a
2975:Vandermonde determinant
2857:characteristic equation
1171:differentiable function
960:Such an equation is an
577:Variation of parameters
567:Separation of variables
356:Peano existence theorem
9867:Differential equations
9836:Carl David Tolmé Runge
9410:Differential-algebraic
9369:Differential equations
8922:
8868:
8714:Cauchy–Euler equations
8642:
8472:
8409:
8380:
8243:
8166:, then one may choose
8158:
8095:
8052:
8006:
7943:
7906:
7877:
7571:unknown functions and
7545:
7489:
7447:
7397:
7310:
7244:
7206:
7147:
7103:
7050:
6995:
6959:
6908:
6844:
6754:
6677:
6591:
6509:
6405:
6358:
6312:
6233:
6026:
5887:
5631:
5516:
5075:
5000:
4878:variation of constants
4755:
4484:
4382:
4260:
4167:
4033:
3988:
3923:
3865:
3807:
3752:
3658:
3610:
3481:
3389:
3332:
3259:
3160:
3080:
2953:
2849:
2757:must be a root of the
2741:
2557:
2428:. It follows that the
2371:
2163:
1959:
1918:
1760:
1500:
1344:
1228:
1114:homogeneous polynomial
886:
646:Carl David Tolmé Runge
189:Differential-algebraic
30:Differential equations
9821:Augustin-Louis Cauchy
9816:Joseph-Louis Lagrange
9710:Finite element method
9700:Crank–Nicolson method
9634:Numerical integration
9613:Exponential stability
9505:Relation to processes
9390:Differential operator
9146:completing the square
9135:Gershenfeld 1999, p.9
9050:radius of convergence
8989:inverse trigonometric
8923:
8869:
8709:Cauchy–Euler equation
8665:Picard–Vessiot theory
8643:
8473:
8410:
8381:
8244:
8159:
8096:
8053:
8007:
7944:
7907:
7905:{\displaystyle b_{n}}
7878:
7546:
7490:
7448:
7398:
7311:
7245:
7207:
7148:
7104:
7051:
6996:
6960:
6909:
6845:
6780:Solving the equation
6755:
6678:
6592:
6510:
6406:
6359:
6313:
6234:
6079:. The computation of
6027:
5888:
5632:
5517:
5076:
5001:
4756:
4493:Finding the solution
4485:
4383:
4261:
4168:
4034:
3989:
3924:
3866:
3808:
3753:
3677:real-valued functions
3659:
3611:
3482:
3390:
3333:
3260:
3161:
3081:
2973:, by considering the
2954:
2850:
2742:
2558:
2402:, who introduced the
2372:
2262:constant coefficients
2164:
1960:
1919:
1761:
1501:
1345:
1229:
1157:Differential operator
1126:constant coefficients
968:may also be a linear
887:
729:that is defined by a
727:differential equation
636:Augustin-Louis Cauchy
621:Joseph-Louis Lagrange
453:Numerical integration
435:Exponential stability
298:Relation to processes
9715:Finite volume method
9639:Dirac delta function
9608:Asymptotic stability
9550:Existence/uniqueness
9415:Integro-differential
8969:exponential function
8880:
8720:
8684:Abel–Ruffini theorem
8482:
8426:
8390:
8268:
8182:
8131:
8094:{\displaystyle U(x)}
8076:
8021:
7967:
7920:
7889:
7583:
7503:
7461:
7411:
7346:
7254:
7216:
7157:
7112:
7060:
7007:
6969:
6918:
6854:
6784:
6687:
6607:
6521:
6437:
6376:
6330:
6256:
6164:
6042:linear equations in
5921:
5652:
5534:
5145:
5010:
4887:
4594:
4396:
4293:
4194:
4108:
3998:
3941:
3875:
3817:
3762:
3707:
3630:
3501:
3453:
3356:
3269:
3185:
3090:
3011:
2971:linearly independent
2862:
2764:
2598:
2454:
2404:exponential function
2268:
2064:
1928:
1770:
1631:
1541:of the operator (if
1377:
1242:
1192:
1065:asymptotic expansion
1021:exponential function
741:
458:Dirac delta function
194:Integro-differential
9725:Perturbation theory
9705:Runge–Kutta methods
9685:Integral transforms
9618:Rate of convergence
9514:(discrete analogue)
9236:Zeilberger, Doron.
9085:asymptotic behavior
9038:recurrence relation
8961:algebraic functions
8934:Holonomic functions
8703:Kovacic's algorithm
8591:
8260:Knowing the matrix
7748:
7602:
7518:
6018:
5991:
5969:
5942:
5879:
5852:
5830:
5803:
5787:
5760:
5744:
5704:
5626:
5586:
5504:
5477:
5455:
5428:
5412:
5385:
5346:
5333:
5311:
5298:
5282:
5269:
5232:
5200:
5174:
3448:times the operator
2264:if it has the form
2191:, if the functions
1576:is usually denoted
1336:
1308:
1182:partial derivatives
1100:algebraic equations
1077:order of derivation
1005:holonomic functions
994:Kovacic's algorithm
974:partial derivatives
554:Perturbation theory
549:Integral transforms
440:Rate of convergence
306:(discrete analogue)
143:Population dynamics
110:Continuum mechanics
101:Applied mathematics
9846:Sofya Kovalevskaya
9680:Integrating factor
9603:Lyapunov stability
9523:Stochastic partial
9079:, localization of
9034:holonomic sequence
9012:closure properties
8946:holonomic function
8940:holonomic function
8918:
8864:
8688:algebraic equation
8638:
8570:
8468:
8405:
8376:
8239:
8154:
8153:
8122:commutes with its
8091:
8048:
8002:
7939:
7902:
7873:
7871:
7736:
7590:
7541:
7506:
7485:
7443:
7393:
7306:
7240:
7202:
7143:
7099:
7046:
6991:
6955:
6904:
6840:
6750:
6673:
6587:
6561:
6505:
6401:
6354:
6353:
6308:
6229:
6022:
5992:
5979:
5943:
5930:
5883:
5853:
5840:
5804:
5791:
5761:
5748:
5724:
5684:
5627:
5606:
5566:
5512:
5510:
5478:
5465:
5429:
5416:
5386:
5373:
5334:
5321:
5299:
5286:
5270:
5257:
5220:
5188:
5162:
5071:
4996:
4871:holonomic function
4863:annihilator method
4751:
4480:
4378:
4256:
4163:
4029:
3984:
3919:
3861:
3803:
3748:
3654:
3606:
3477:
3385:
3328:
3255:
3156:
3076:
2949:
2845:
2737:
2553:
2367:
2213:are continuous in
2159:
1955:
1914:
1756:
1496:
1371:linear combination
1340:
1315:
1287:
1224:
1130:constant functions
882:
544:Integrating factor
385:Initial conditions
320:Stochastic partial
9854:
9853:
9733:
9732:
9538:
9537:
9295:978-0-521-57095-4
9222:, as in the text.
9107:Laplace transform
9102:Fourier transform
9073:definite integral
8997:special functions
8985:hyperbolic cosine
8950:D-finite function
8948:, also called a
8198:
7951:are functions of
7301:
6986:
6950:
6934:
6896:
6826:
6623:
6560:
6272:
4277:complex conjugate
3933:Second-order case
3646:
3522:
3469:
3342:
3341:
2432:th derivative of
1924:may be rewritten
1751:
1694:
1491:
1434:
1338:
1222:
1071:Basic terminology
1017:special functions
980:Types of solution
731:linear polynomial
715:
714:
606:Gottfried Leibniz
497:Finite difference
289:
288:
150:
149:
120:Dynamical systems
9874:
9831:Phyllis Nicolson
9811:Rudolf Lipschitz
9648:Solution methods
9623:Series solutions
9547:
9546:
9380:
9379:
9362:
9355:
9348:
9339:
9338:
9316:
9298:
9280:
9258:
9249:
9243:
9234:
9223:
9221:
9207:
9197:
9182:
9176:
9161:
9142:
9136:
9133:
9001:Bessel functions
8929:
8927:
8925:
8924:
8919:
8917:
8916:
8892:
8891:
8873:
8871:
8870:
8865:
8842:
8841:
8814:
8813:
8792:
8791:
8776:
8775:
8748:
8747:
8732:
8731:
8647:
8645:
8644:
8639:
8618:
8604:
8603:
8590:
8585:
8584:
8583:
8554:
8553:
8552:
8539:
8538:
8526:
8525:
8489:
8477:
8475:
8474:
8469:
8464:
8463:
8458:
8446:
8445:
8433:
8415:is an arbitrary
8414:
8412:
8411:
8406:
8404:
8403:
8402:
8385:
8383:
8382:
8377:
8356:
8342:
8341:
8311:
8310:
8309:
8275:
8263:
8255:Magnus expansion
8251:numerical method
8248:
8246:
8245:
8240:
8199:
8197:
8186:
8177:
8169:
8165:
8163:
8161:
8160:
8155:
8121:
8117:
8113:
8102:
8100:
8098:
8097:
8092:
8065:
8057:
8055:
8054:
8049:
8044:
8033:
8029:
8011:
8009:
8008:
8003:
7998:
7990:
7979:
7975:
7962:
7954:
7950:
7948:
7946:
7945:
7940:
7938:
7937:
7913:
7911:
7909:
7908:
7903:
7901:
7900:
7882:
7880:
7879:
7874:
7872:
7865:
7864:
7846:
7845:
7821:
7820:
7802:
7801:
7774:
7773:
7744:
7723:
7719:
7718:
7700:
7699:
7675:
7674:
7656:
7655:
7628:
7627:
7598:
7574:
7570:
7563:
7552:
7550:
7548:
7547:
7542:
7537:
7536:
7514:
7496:
7494:
7492:
7491:
7486:
7484:
7483:
7471:
7454:
7452:
7450:
7449:
7444:
7442:
7441:
7423:
7422:
7404:
7402:
7400:
7399:
7394:
7392:
7391:
7367:
7356:
7315:
7313:
7312:
7307:
7302:
7297:
7286:
7281:
7280:
7249:
7247:
7246:
7241:
7211:
7209:
7208:
7203:
7195:
7184:
7183:
7152:
7150:
7149:
7144:
7133:
7132:
7108:
7106:
7105:
7100:
7095:
7094:
7079:
7055:
7053:
7052:
7047:
7042:
7041:
7020:
7000:
6998:
6997:
6992:
6987:
6979:
6964:
6962:
6961:
6956:
6951:
6943:
6935:
6930:
6922:
6913:
6911:
6910:
6905:
6897:
6892:
6878:
6864:
6849:
6847:
6846:
6841:
6827:
6822:
6808:
6794:
6771:
6767:
6763:
6759:
6757:
6756:
6751:
6740:
6739:
6721:
6720:
6708:
6707:
6682:
6680:
6679:
6674:
6669:
6668:
6650:
6646:
6645:
6644:
6624:
6622:
6611:
6598:
6596:
6594:
6593:
6588:
6583:
6579:
6578:
6562:
6559:
6548:
6542:
6541:
6514:
6512:
6511:
6506:
6501:
6500:
6482:
6481:
6460:
6459:
6447:
6432:
6420:
6410:
6408:
6407:
6402:
6397:
6396:
6371:
6363:
6361:
6360:
6355:
6322:is an arbitrary
6321:
6317:
6315:
6314:
6309:
6273:
6268:
6260:
6251:
6238:
6236:
6235:
6230:
6174:
6159:
6136:
6100:
6074:
6065:
6061:
6041:
6037:
6031:
6029:
6028:
6023:
6017:
6000:
5987:
5968:
5951:
5938:
5916:
5898:
5892:
5890:
5889:
5884:
5878:
5861:
5848:
5829:
5812:
5799:
5786:
5769:
5756:
5743:
5732:
5723:
5722:
5703:
5692:
5683:
5682:
5670:
5669:
5647:
5636:
5634:
5633:
5628:
5625:
5614:
5605:
5604:
5585:
5574:
5565:
5564:
5552:
5551:
5522:which imply (by
5521:
5519:
5518:
5513:
5511:
5503:
5486:
5473:
5454:
5437:
5424:
5411:
5394:
5381:
5350:
5342:
5329:
5307:
5294:
5278:
5265:
5242:
5241:
5228:
5210:
5209:
5196:
5184:
5183:
5170:
5140:
5136:
5118:
5100:
5080:
5078:
5077:
5072:
5067:
5066:
5057:
5056:
5038:
5037:
5028:
5027:
5005:
5003:
5002:
4997:
4986:
4985:
4973:
4965:
4964:
4940:
4939:
4918:
4917:
4905:
4904:
4868:
4856:
4852:
4848:
4837:
4826:
4817:
4809:
4805:
4798:
4790:
4786:
4782:
4778:
4760:
4758:
4757:
4752:
4720:
4719:
4698:
4690:
4689:
4656:
4655:
4634:
4633:
4612:
4611:
4589:
4577:
4569:
4560:
4551:
4542:
4533:
4529:
4516:
4503:
4489:
4487:
4486:
4481:
4455:
4454:
4424:
4423:
4411:
4410:
4387:
4385:
4384:
4379:
4374:
4373:
4346:
4345:
4333:
4332:
4305:
4304:
4288:
4274:
4265:
4263:
4262:
4257:
4252:
4251:
4247:
4222:
4221:
4209:
4208:
4189:
4181:
4172:
4170:
4169:
4164:
4159:
4158:
4146:
4145:
4133:
4132:
4120:
4119:
4103:
4099:
4095:
4084:
4075:
4066:
4048:
4044:
4038:
4036:
4035:
4030:
4010:
4009:
3993:
3991:
3990:
3985:
3965:
3951:
3928:
3926:
3925:
3920:
3900:
3899:
3887:
3886:
3870:
3868:
3867:
3862:
3842:
3841:
3829:
3828:
3812:
3810:
3809:
3804:
3802:
3801:
3774:
3773:
3757:
3755:
3754:
3749:
3747:
3746:
3719:
3718:
3703:, and replacing
3698:
3688:
3665:
3663:
3661:
3660:
3655:
3647:
3645:
3634:
3624:
3615:
3613:
3612:
3607:
3602:
3601:
3589:
3588:
3567:
3563:
3562:
3561:
3549:
3548:
3534:
3530:
3523:
3521:
3510:
3492:
3488:
3486:
3484:
3483:
3478:
3470:
3468:
3457:
3447:
3443:
3424:
3420:
3416:
3406:
3402:
3398:
3394:
3392:
3391:
3386:
3381:
3380:
3368:
3367:
3337:
3335:
3334:
3329:
3324:
3323:
3307:
3306:
3264:
3262:
3261:
3256:
3251:
3250:
3234:
3233:
3220:
3219:
3200:
3199:
3180:
3176:
3169:
3166:This has zeros,
3165:
3163:
3162:
3157:
3134:
3133:
3118:
3117:
3102:
3101:
3085:
3083:
3082:
3077:
3063:
3049:
3035:
3021:
2999:
2998:
2987:
2968:
2958:
2956:
2955:
2950:
2942:
2941:
2932:
2931:
2913:
2912:
2903:
2902:
2887:
2886:
2874:
2873:
2854:
2852:
2851:
2846:
2844:
2843:
2834:
2833:
2815:
2814:
2805:
2804:
2789:
2788:
2776:
2775:
2756:
2752:
2746:
2744:
2743:
2738:
2730:
2729:
2717:
2716:
2707:
2706:
2688:
2687:
2675:
2674:
2665:
2664:
2652:
2651:
2636:
2635:
2623:
2622:
2610:
2609:
2593:
2589:
2580:
2562:
2560:
2559:
2554:
2546:
2545:
2530:
2529:
2511:
2503:
2502:
2490:
2482:
2481:
2466:
2465:
2446:
2437:
2431:
2427:
2420:
2410:
2394:
2376:
2374:
2373:
2368:
2360:
2359:
2344:
2343:
2325:
2317:
2316:
2304:
2296:
2295:
2280:
2279:
2251:
2247:
2243:
2238:
2220:
2216:
2212:
2190:
2186:
2168:
2166:
2165:
2160:
2146:
2145:
2136:
2135:
2108:
2107:
2098:
2097:
2076:
2075:
2059:
2040:
2036:
2028:
2021:
1999:
1989:
1970:
1964:
1962:
1961:
1956:
1923:
1921:
1920:
1915:
1898:
1897:
1873:
1872:
1854:
1837:
1836:
1824:
1807:
1806:
1782:
1781:
1765:
1763:
1762:
1757:
1752:
1750:
1749:
1748:
1735:
1734:
1725:
1714:
1713:
1695:
1693:
1682:
1671:
1670:
1649:
1648:
1592:
1581:
1575:
1571:
1567:
1556:
1536:
1532:
1505:
1503:
1502:
1497:
1492:
1490:
1489:
1488:
1475:
1474:
1465:
1454:
1453:
1435:
1433:
1422:
1411:
1410:
1389:
1388:
1353:
1349:
1347:
1346:
1341:
1339:
1337:
1335:
1334:
1333:
1323:
1307:
1306:
1305:
1295:
1282:
1281:
1280:
1279:
1261:
1260:
1246:
1238:functions, and
1233:
1231:
1230:
1225:
1223:
1221:
1220:
1219:
1206:
1205:
1196:
1187:
1177:
1168:
1141:
1140:
1122:
1121:
1093:
1045:Bessel functions
956:
953:of the variable
952:
948:
934:
923:
922:
906:
891:
889:
888:
883:
866:
865:
841:
840:
825:
808:
807:
795:
778:
777:
753:
752:
707:
700:
693:
671:Phyllis Nicolson
656:Rudolf Lipschitz
539:Green's function
515:Infinite element
506:
471:Solution methods
449:
307:
218:By variable type
172:
171:
54:Natural sciences
47:
46:
26:
25:
9882:
9881:
9877:
9876:
9875:
9873:
9872:
9871:
9857:
9856:
9855:
9850:
9791:Jacob Bernoulli
9764:
9729:
9720:Galerkin method
9643:
9581:Solution topics
9576:
9534:
9500:
9439:
9371:
9366:
9323:
9314:
9296:
9278:
9262:
9261:
9250:
9246:
9235:
9226:
9209:
9199:
9184:
9178:
9163:
9149:
9143:
9139:
9134:
9130:
9125:
9093:
8981:hyperbolic sine
8942:
8936:
8906:
8902:
8887:
8883:
8881:
8878:
8877:
8875:
8837:
8833:
8797:
8793:
8781:
8777:
8765:
8761:
8737:
8733:
8727:
8723:
8721:
8718:
8717:
8711:
8653:
8614:
8596:
8592:
8586:
8579:
8575:
8574:
8548:
8544:
8543:
8534:
8530:
8518:
8514:
8485:
8483:
8480:
8479:
8459:
8454:
8453:
8441:
8437:
8429:
8427:
8424:
8423:
8398:
8394:
8393:
8391:
8388:
8387:
8352:
8334:
8330:
8305:
8301:
8300:
8271:
8269:
8266:
8265:
8261:
8190:
8185:
8183:
8180:
8179:
8175:
8167:
8132:
8129:
8128:
8126:
8119:
8115:
8108:
8077:
8074:
8073:
8071:
8063:
8040:
8025:
8024:
8022:
8019:
8018:
7994:
7986:
7971:
7970:
7968:
7965:
7964:
7956:
7952:
7927:
7923:
7921:
7918:
7917:
7915:
7896:
7892:
7890:
7887:
7886:
7884:
7870:
7869:
7860:
7856:
7835:
7831:
7816:
7812:
7791:
7787:
7769:
7765:
7758:
7740:
7733:
7732:
7721:
7720:
7714:
7710:
7689:
7685:
7670:
7666:
7645:
7641:
7623:
7619:
7612:
7594:
7586:
7584:
7581:
7580:
7572:
7568:
7554:
7526:
7522:
7510:
7504:
7501:
7500:
7498:
7479:
7475:
7464:
7462:
7459:
7458:
7456:
7437:
7433:
7418:
7414:
7412:
7409:
7408:
7406:
7381:
7377:
7360:
7349:
7347:
7344:
7343:
7341:
7333:
7327:
7321:
7287:
7285:
7276:
7272:
7255:
7252:
7251:
7217:
7214:
7213:
7191:
7179:
7175:
7158:
7155:
7154:
7128:
7124:
7113:
7110:
7109:
7090:
7086:
7072:
7061:
7058:
7057:
7037:
7033:
7013:
7008:
7005:
7004:
6978:
6970:
6967:
6966:
6942:
6923:
6921:
6919:
6916:
6915:
6879:
6877:
6857:
6855:
6852:
6851:
6809:
6807:
6787:
6785:
6782:
6781:
6778:
6769:
6765:
6761:
6732:
6728:
6716:
6712:
6703:
6699:
6688:
6685:
6684:
6661:
6657:
6637:
6633:
6629:
6625:
6615:
6610:
6608:
6605:
6604:
6571:
6567:
6563:
6552:
6546:
6534:
6530:
6522:
6519:
6518:
6516:
6493:
6489:
6474:
6470:
6452:
6448:
6440:
6438:
6435:
6434:
6428:
6412:
6392:
6388:
6377:
6374:
6373:
6369:
6331:
6328:
6327:
6319:
6261:
6259:
6257:
6254:
6253:
6242:
6167:
6165:
6162:
6161:
6150:
6147:
6135:
6127:
6118:
6112:
6102:
6099:
6090:
6084:
6081:antiderivatives
6073:
6067:
6063:
6060:
6050:
6043:
6039:
6035:
6001:
5996:
5983:
5952:
5947:
5934:
5922:
5919:
5918:
5915:
5906:
5900:
5896:
5862:
5857:
5844:
5813:
5808:
5795:
5770:
5765:
5752:
5733:
5728:
5718:
5714:
5693:
5688:
5678:
5674:
5659:
5655:
5653:
5650:
5649:
5638:
5615:
5610:
5600:
5596:
5575:
5570:
5560:
5556:
5541:
5537:
5535:
5532:
5531:
5509:
5508:
5487:
5482:
5469:
5438:
5433:
5420:
5395:
5390:
5377:
5366:
5360:
5359:
5348:
5347:
5338:
5325:
5303:
5290:
5274:
5261:
5250:
5244:
5243:
5237:
5233:
5224:
5205:
5201:
5192:
5179:
5175:
5166:
5155:
5148:
5146:
5143:
5142:
5138:
5135:
5126:
5120:
5117:
5108:
5102:
5098:
5089:
5082:
5062:
5058:
5052:
5048:
5033:
5029:
5023:
5019:
5011:
5008:
5007:
4981:
4977:
4966:
4954:
4950:
4923:
4919:
4913:
4909:
4894:
4890:
4888:
4885:
4884:
4866:
4854:
4850:
4839:
4828:
4819:
4815:
4807:
4803:
4792:
4788:
4784:
4780:
4777:
4768:
4762:
4715:
4711:
4691:
4679:
4675:
4639:
4635:
4629:
4625:
4601:
4597:
4595:
4592:
4591:
4587:
4584:
4575:
4568:
4562:
4559:
4553:
4550:
4544:
4541:
4535:
4531:
4528:
4518:
4515:
4505:
4494:
4450:
4446:
4419:
4415:
4403:
4399:
4397:
4394:
4393:
4390:Euler's formula
4351:
4347:
4341:
4337:
4310:
4306:
4300:
4296:
4294:
4291:
4290:
4280:
4269:
4243:
4233:
4229:
4217:
4213:
4204:
4200:
4195:
4192:
4191:
4183:
4176:
4151:
4147:
4141:
4137:
4125:
4121:
4115:
4111:
4109:
4106:
4105:
4101:
4097:
4090:
4083:
4077:
4074:
4068:
4054:
4046:
4042:
4005:
4001:
3999:
3996:
3995:
3958:
3944:
3942:
3939:
3938:
3935:
3892:
3888:
3882:
3878:
3876:
3873:
3872:
3834:
3830:
3824:
3820:
3818:
3815:
3814:
3779:
3775:
3769:
3765:
3763:
3760:
3759:
3724:
3720:
3714:
3710:
3708:
3705:
3704:
3701:Euler's formula
3690:
3680:
3638:
3633:
3631:
3628:
3627:
3626:
3625:application of
3619:
3594:
3590:
3578:
3574:
3554:
3550:
3544:
3540:
3539:
3535:
3514:
3509:
3508:
3504:
3502:
3499:
3498:
3490:
3461:
3456:
3454:
3451:
3450:
3449:
3445:
3426:
3422:
3418:
3408:
3404:
3400:
3396:
3373:
3369:
3363:
3359:
3357:
3354:
3353:
3319:
3315:
3302:
3298:
3270:
3267:
3266:
3246:
3242:
3229:
3225:
3209:
3205:
3192:
3188:
3186:
3183:
3182:
3178:
3171:
3167:
3129:
3125:
3113:
3109:
3097:
3093:
3091:
3088:
3087:
3056:
3042:
3028:
3014:
3012:
3009:
3008:
2978:
2966:
2937:
2933:
2927:
2923:
2908:
2904:
2898:
2894:
2882:
2878:
2869:
2865:
2863:
2860:
2859:
2839:
2835:
2829:
2825:
2810:
2806:
2800:
2796:
2784:
2780:
2771:
2767:
2765:
2762:
2761:
2754:
2748:
2722:
2718:
2712:
2708:
2702:
2698:
2680:
2676:
2670:
2666:
2660:
2656:
2644:
2640:
2631:
2627:
2615:
2611:
2605:
2601:
2599:
2596:
2595:
2591:
2585:
2579:
2570:
2564:
2535:
2531:
2525:
2521:
2504:
2498:
2494:
2483:
2477:
2473:
2461:
2457:
2455:
2452:
2451:
2439:
2433:
2429:
2422:
2412:
2406:
2393:
2384:
2378:
2349:
2345:
2339:
2335:
2318:
2312:
2308:
2297:
2291:
2287:
2275:
2271:
2269:
2266:
2265:
2258:
2249:
2245:
2232:
2224:
2222:
2218:
2214:
2211:
2202:
2192:
2188:
2185:
2176:
2170:
2141:
2137:
2131:
2127:
2103:
2099:
2093:
2089:
2071:
2067:
2065:
2062:
2061:
2042:
2038:
2034:
2026:
2016:
1991:
1972:
1968:
1929:
1926:
1925:
1887:
1883:
1868:
1864:
1847:
1832:
1828:
1817:
1802:
1798:
1777:
1773:
1771:
1768:
1767:
1744:
1740:
1736:
1730:
1726:
1724:
1709:
1705:
1686:
1681:
1666:
1662:
1644:
1640:
1632:
1629:
1628:
1614:complex numbers
1595:linear operator
1583:
1577:
1573:
1569:
1565:
1550:
1542:
1534:
1526:
1513:
1507:
1484:
1480:
1476:
1470:
1466:
1464:
1449:
1445:
1426:
1421:
1406:
1402:
1384:
1380:
1378:
1375:
1374:
1363:linear operator
1351:
1329:
1325:
1324:
1319:
1301:
1297:
1296:
1291:
1283:
1275:
1271:
1256:
1252:
1251:
1247:
1245:
1243:
1240:
1239:
1234:in the case of
1215:
1211:
1207:
1201:
1197:
1195:
1193:
1190:
1189:
1185:
1175:
1166:
1159:
1153:
1138:
1137:
1119:
1118:
1084:
1073:
1057:antiderivatives
1009:differentiation
982:
954:
950:
940:
925:
916:
908:
900:
894:
893:
855:
851:
836:
832:
818:
803:
799:
788:
773:
769:
748:
744:
742:
739:
738:
711:
682:
681:
680:
611:Jacob Bernoulli
595:
582:
581:
563:
532:Petrov–Galerkin
500:
485:
472:
464:
463:
462:
444:
390:Boundary values
379:
371:
370:
346:
333:
332:
331:
305:
299:
291:
290:
278:
255:
213:
169:
156:
155:
151:
129:Social sciences
85:
63:
44:
24:
17:
12:
11:
5:
9880:
9870:
9869:
9852:
9851:
9849:
9848:
9843:
9838:
9833:
9828:
9823:
9818:
9813:
9808:
9806:Ernst Lindelöf
9803:
9798:
9793:
9788:
9786:Leonhard Euler
9783:
9778:
9772:
9770:
9769:Mathematicians
9766:
9765:
9763:
9762:
9757:
9752:
9747:
9741:
9739:
9735:
9734:
9731:
9730:
9728:
9727:
9722:
9717:
9712:
9707:
9702:
9697:
9692:
9687:
9682:
9677:
9672:
9667:
9662:
9657:
9651:
9649:
9645:
9644:
9642:
9641:
9636:
9631:
9625:
9620:
9615:
9610:
9605:
9600:
9595:
9593:Phase portrait
9590:
9584:
9582:
9578:
9577:
9575:
9574:
9569:
9564:
9559:
9553:
9551:
9544:
9540:
9539:
9536:
9535:
9533:
9532:
9527:
9526:
9525:
9515:
9508:
9506:
9502:
9501:
9499:
9498:
9496:On jet bundles
9493:
9488:
9483:
9478:
9473:
9468:
9463:
9461:Nonhomogeneous
9458:
9453:
9447:
9445:
9441:
9440:
9438:
9437:
9432:
9427:
9422:
9417:
9412:
9407:
9402:
9397:
9392:
9386:
9384:
9377:
9376:Classification
9373:
9372:
9365:
9364:
9357:
9350:
9342:
9336:
9335:
9329:
9322:
9321:External links
9319:
9318:
9317:
9312:
9299:
9294:
9281:
9276:
9260:
9259:
9244:
9224:
9137:
9127:
9126:
9124:
9121:
9120:
9119:
9114:
9109:
9104:
9099:
9092:
9089:
8938:Main article:
8935:
8932:
8915:
8912:
8909:
8905:
8901:
8898:
8895:
8890:
8886:
8863:
8860:
8857:
8854:
8851:
8848:
8845:
8840:
8836:
8832:
8829:
8826:
8823:
8820:
8817:
8812:
8809:
8806:
8803:
8800:
8796:
8790:
8787:
8784:
8780:
8774:
8771:
8768:
8764:
8760:
8757:
8754:
8751:
8746:
8743:
8740:
8736:
8730:
8726:
8710:
8707:
8673:Ernest Vessiot
8652:
8649:
8637:
8634:
8631:
8627:
8624:
8621:
8617:
8613:
8610:
8607:
8602:
8599:
8595:
8589:
8582:
8578:
8573:
8569:
8566:
8563:
8560:
8557:
8551:
8547:
8542:
8537:
8533:
8529:
8524:
8521:
8517:
8513:
8510:
8507:
8504:
8501:
8498:
8495:
8492:
8488:
8467:
8462:
8457:
8452:
8449:
8444:
8440:
8436:
8432:
8401:
8397:
8375:
8372:
8369:
8365:
8362:
8359:
8355:
8351:
8348:
8345:
8340:
8337:
8333:
8329:
8326:
8323:
8320:
8317:
8314:
8308:
8304:
8299:
8296:
8293:
8290:
8287:
8284:
8281:
8278:
8274:
8238:
8235:
8232:
8229:
8226:
8223:
8220:
8217:
8214:
8211:
8208:
8205:
8202:
8196:
8193:
8189:
8152:
8149:
8146:
8143:
8140:
8137:
8124:antiderivative
8090:
8087:
8084:
8081:
8047:
8043:
8039:
8036:
8032:
8028:
8001:
7997:
7993:
7989:
7985:
7982:
7978:
7974:
7936:
7933:
7930:
7926:
7899:
7895:
7868:
7863:
7859:
7855:
7852:
7849:
7844:
7841:
7838:
7834:
7830:
7827:
7824:
7819:
7815:
7811:
7808:
7805:
7800:
7797:
7794:
7790:
7786:
7783:
7780:
7777:
7772:
7768:
7764:
7761:
7759:
7757:
7754:
7751:
7747:
7743:
7739:
7735:
7734:
7731:
7726:
7724:
7722:
7717:
7713:
7709:
7706:
7703:
7698:
7695:
7692:
7688:
7684:
7681:
7678:
7673:
7669:
7665:
7662:
7659:
7654:
7651:
7648:
7644:
7640:
7637:
7634:
7631:
7626:
7622:
7618:
7615:
7613:
7611:
7608:
7605:
7601:
7597:
7593:
7589:
7588:
7540:
7535:
7532:
7529:
7525:
7521:
7517:
7513:
7509:
7482:
7478:
7474:
7470:
7467:
7440:
7436:
7432:
7429:
7426:
7421:
7417:
7390:
7387:
7384:
7380:
7376:
7373:
7370:
7366:
7363:
7359:
7355:
7352:
7323:Main article:
7320:
7317:
7305:
7300:
7296:
7293:
7290:
7284:
7279:
7275:
7271:
7268:
7265:
7262:
7259:
7239:
7236:
7233:
7230:
7227:
7224:
7221:
7201:
7198:
7194:
7190:
7187:
7182:
7178:
7174:
7171:
7168:
7165:
7162:
7142:
7139:
7136:
7131:
7127:
7123:
7120:
7117:
7098:
7093:
7089:
7085:
7082:
7078:
7075:
7071:
7068:
7065:
7045:
7040:
7036:
7032:
7029:
7026:
7023:
7019:
7016:
7012:
6990:
6985:
6982:
6977:
6974:
6954:
6949:
6946:
6941:
6938:
6933:
6929:
6926:
6903:
6900:
6895:
6891:
6888:
6885:
6882:
6876:
6873:
6870:
6867:
6863:
6860:
6839:
6836:
6833:
6830:
6825:
6821:
6818:
6815:
6812:
6806:
6803:
6800:
6797:
6793:
6790:
6777:
6774:
6749:
6746:
6743:
6738:
6735:
6731:
6727:
6724:
6719:
6715:
6711:
6706:
6702:
6698:
6695:
6692:
6672:
6667:
6664:
6660:
6656:
6653:
6649:
6643:
6640:
6636:
6632:
6628:
6621:
6618:
6614:
6586:
6582:
6577:
6574:
6570:
6566:
6558:
6555:
6551:
6545:
6540:
6537:
6533:
6529:
6526:
6504:
6499:
6496:
6492:
6488:
6485:
6480:
6477:
6473:
6469:
6466:
6463:
6458:
6455:
6451:
6446:
6443:
6400:
6395:
6391:
6387:
6384:
6381:
6366:antiderivative
6352:
6349:
6345:
6342:
6338:
6335:
6307:
6304:
6301:
6298:
6295:
6292:
6289:
6286:
6282:
6279:
6276:
6271:
6267:
6264:
6228:
6225:
6222:
6219:
6216:
6213:
6210:
6207:
6204:
6201:
6198:
6195:
6192:
6189:
6186:
6183:
6180:
6177:
6173:
6170:
6146:
6143:
6131:
6123:
6116:
6110:
6095:
6088:
6077:linear algebra
6071:
6056:
6048:
6021:
6016:
6013:
6010:
6007:
6004:
5999:
5995:
5990:
5986:
5982:
5978:
5975:
5972:
5967:
5964:
5961:
5958:
5955:
5950:
5946:
5941:
5937:
5933:
5929:
5926:
5911:
5904:
5882:
5877:
5874:
5871:
5868:
5865:
5860:
5856:
5851:
5847:
5843:
5839:
5836:
5833:
5828:
5825:
5822:
5819:
5816:
5811:
5807:
5802:
5798:
5794:
5790:
5785:
5782:
5779:
5776:
5773:
5768:
5764:
5759:
5755:
5751:
5747:
5742:
5739:
5736:
5731:
5727:
5721:
5717:
5713:
5710:
5707:
5702:
5699:
5696:
5691:
5687:
5681:
5677:
5673:
5668:
5665:
5662:
5658:
5624:
5621:
5618:
5613:
5609:
5603:
5599:
5595:
5592:
5589:
5584:
5581:
5578:
5573:
5569:
5563:
5559:
5555:
5550:
5547:
5544:
5540:
5507:
5502:
5499:
5496:
5493:
5490:
5485:
5481:
5476:
5472:
5468:
5464:
5461:
5458:
5453:
5450:
5447:
5444:
5441:
5436:
5432:
5427:
5423:
5419:
5415:
5410:
5407:
5404:
5401:
5398:
5393:
5389:
5384:
5380:
5376:
5372:
5369:
5367:
5365:
5362:
5361:
5358:
5353:
5351:
5349:
5345:
5341:
5337:
5332:
5328:
5324:
5320:
5317:
5314:
5310:
5306:
5302:
5297:
5293:
5289:
5285:
5281:
5277:
5273:
5268:
5264:
5260:
5256:
5253:
5251:
5249:
5246:
5245:
5240:
5236:
5231:
5227:
5223:
5219:
5216:
5213:
5208:
5204:
5199:
5195:
5191:
5187:
5182:
5178:
5173:
5169:
5165:
5161:
5158:
5156:
5154:
5151:
5150:
5131:
5124:
5113:
5106:
5094:
5087:
5070:
5065:
5061:
5055:
5051:
5047:
5044:
5041:
5036:
5032:
5026:
5022:
5018:
5015:
4995:
4992:
4989:
4984:
4980:
4976:
4972:
4969:
4963:
4960:
4957:
4953:
4949:
4946:
4943:
4938:
4935:
4932:
4929:
4926:
4922:
4916:
4912:
4908:
4903:
4900:
4897:
4893:
4773:
4766:
4750:
4747:
4744:
4741:
4738:
4735:
4732:
4729:
4726:
4723:
4718:
4714:
4710:
4707:
4704:
4701:
4697:
4694:
4688:
4685:
4682:
4678:
4674:
4671:
4668:
4665:
4662:
4659:
4654:
4651:
4648:
4645:
4642:
4638:
4632:
4628:
4624:
4621:
4618:
4615:
4610:
4607:
4604:
4600:
4583:
4580:
4572:Cauchy problem
4566:
4557:
4548:
4539:
4526:
4513:
4491:
4490:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4453:
4449:
4445:
4442:
4439:
4436:
4433:
4430:
4427:
4422:
4418:
4414:
4409:
4406:
4402:
4377:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4350:
4344:
4340:
4336:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4309:
4303:
4299:
4266:
4255:
4250:
4246:
4242:
4239:
4236:
4232:
4228:
4225:
4220:
4216:
4212:
4207:
4203:
4199:
4173:
4162:
4157:
4154:
4150:
4144:
4140:
4136:
4131:
4128:
4124:
4118:
4114:
4081:
4072:
4028:
4025:
4022:
4019:
4016:
4013:
4008:
4004:
3983:
3980:
3977:
3974:
3971:
3968:
3964:
3961:
3957:
3954:
3950:
3947:
3934:
3931:
3918:
3915:
3912:
3909:
3906:
3903:
3898:
3895:
3891:
3885:
3881:
3860:
3857:
3854:
3851:
3848:
3845:
3840:
3837:
3833:
3827:
3823:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3778:
3772:
3768:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3723:
3717:
3713:
3653:
3650:
3644:
3641:
3637:
3605:
3600:
3597:
3593:
3587:
3584:
3581:
3577:
3573:
3570:
3566:
3560:
3557:
3553:
3547:
3543:
3538:
3533:
3529:
3526:
3520:
3517:
3513:
3507:
3476:
3473:
3467:
3464:
3460:
3384:
3379:
3376:
3372:
3366:
3362:
3350:multiple roots
3340:
3339:
3327:
3322:
3318:
3314:
3310:
3305:
3301:
3296:
3293:
3290:
3287:
3283:
3280:
3277:
3274:
3254:
3249:
3245:
3241:
3237:
3232:
3228:
3223:
3218:
3215:
3212:
3208:
3203:
3198:
3195:
3191:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3132:
3128:
3124:
3121:
3116:
3112:
3108:
3105:
3100:
3096:
3075:
3072:
3069:
3066:
3062:
3059:
3055:
3052:
3048:
3045:
3041:
3038:
3034:
3031:
3027:
3024:
3020:
3017:
3004:
3003:
2948:
2945:
2940:
2936:
2930:
2926:
2922:
2919:
2916:
2911:
2907:
2901:
2897:
2893:
2890:
2885:
2881:
2877:
2872:
2868:
2842:
2838:
2832:
2828:
2824:
2821:
2818:
2813:
2809:
2803:
2799:
2795:
2792:
2787:
2783:
2779:
2774:
2770:
2747:Factoring out
2736:
2733:
2728:
2725:
2721:
2715:
2711:
2705:
2701:
2697:
2694:
2691:
2686:
2683:
2679:
2673:
2669:
2663:
2659:
2655:
2650:
2647:
2643:
2639:
2634:
2630:
2626:
2621:
2618:
2614:
2608:
2604:
2575:
2568:
2552:
2549:
2544:
2541:
2538:
2534:
2528:
2524:
2520:
2517:
2514:
2510:
2507:
2501:
2497:
2493:
2489:
2486:
2480:
2476:
2472:
2469:
2464:
2460:
2400:Leonhard Euler
2389:
2382:
2366:
2363:
2358:
2355:
2352:
2348:
2342:
2338:
2334:
2331:
2328:
2324:
2321:
2315:
2311:
2307:
2303:
2300:
2294:
2290:
2286:
2283:
2278:
2274:
2257:
2254:
2228:
2207:
2200:
2181:
2174:
2158:
2155:
2152:
2149:
2144:
2140:
2134:
2130:
2126:
2123:
2120:
2117:
2114:
2111:
2106:
2102:
2096:
2092:
2088:
2085:
2082:
2079:
2074:
2070:
2060:have the form
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1913:
1910:
1907:
1904:
1901:
1896:
1893:
1890:
1886:
1882:
1879:
1876:
1871:
1867:
1863:
1860:
1857:
1853:
1850:
1846:
1843:
1840:
1835:
1831:
1827:
1823:
1820:
1816:
1813:
1810:
1805:
1801:
1797:
1794:
1791:
1788:
1785:
1780:
1776:
1755:
1747:
1743:
1739:
1733:
1729:
1723:
1720:
1717:
1712:
1708:
1704:
1701:
1698:
1692:
1689:
1685:
1680:
1677:
1674:
1669:
1665:
1661:
1658:
1655:
1652:
1647:
1643:
1639:
1636:
1572:to a function
1546:
1522:
1511:
1495:
1487:
1483:
1479:
1473:
1469:
1463:
1460:
1457:
1452:
1448:
1444:
1441:
1438:
1432:
1429:
1425:
1420:
1417:
1414:
1409:
1405:
1401:
1398:
1395:
1392:
1387:
1383:
1332:
1328:
1322:
1318:
1314:
1311:
1304:
1300:
1294:
1290:
1286:
1278:
1274:
1270:
1267:
1264:
1259:
1255:
1250:
1218:
1214:
1210:
1204:
1200:
1155:Main article:
1152:
1149:
1072:
1069:
1041:error function
981:
978:
935:are arbitrary
912:
898:
881:
878:
875:
872:
869:
864:
861:
858:
854:
850:
847:
844:
839:
835:
831:
828:
824:
821:
817:
814:
811:
806:
802:
798:
794:
791:
787:
784:
781:
776:
772:
768:
765:
762:
759:
756:
751:
747:
713:
712:
710:
709:
702:
695:
687:
684:
683:
679:
678:
673:
668:
663:
661:Ernst Lindelöf
658:
653:
648:
643:
638:
633:
631:Joseph Fourier
628:
623:
618:
616:Leonhard Euler
613:
608:
603:
597:
596:
593:
592:
589:
588:
584:
583:
580:
579:
574:
569:
562:
561:
556:
551:
546:
541:
536:
535:
534:
524:
519:
518:
517:
510:Finite element
507:
503:Crank–Nicolson
494:
489:
483:
478:
474:
473:
470:
469:
466:
465:
461:
460:
455:
450:
442:
437:
424:
422:Phase portrait
419:
414:
413:
412:
410:Cauchy problem
407:
402:
397:
387:
381:
380:
378:General topics
377:
376:
373:
372:
369:
368:
363:
358:
353:
347:
344:
343:
340:
339:
335:
334:
330:
329:
324:
323:
322:
311:
310:
309:
300:
297:
296:
293:
292:
287:
286:
285:
284:
277:
276:
271:
265:
262:
261:
257:
256:
254:
253:
251:Nonhomogeneous
244:
239:
236:
230:
229:
228:
220:
219:
215:
214:
212:
211:
206:
201:
196:
191:
186:
181:
175:
170:
167:
166:
163:
162:
161:Classification
158:
157:
148:
147:
146:
145:
140:
132:
131:
125:
124:
123:
122:
117:
112:
104:
103:
97:
96:
95:
94:
89:
83:
78:
73:
65:
64:
62:
61:
56:
50:
45:
42:
41:
38:
37:
33:
32:
15:
9:
6:
4:
3:
2:
9879:
9868:
9865:
9864:
9862:
9847:
9844:
9842:
9839:
9837:
9834:
9832:
9829:
9827:
9824:
9822:
9819:
9817:
9814:
9812:
9809:
9807:
9804:
9802:
9799:
9797:
9794:
9792:
9789:
9787:
9784:
9782:
9779:
9777:
9774:
9773:
9771:
9767:
9761:
9758:
9756:
9753:
9751:
9748:
9746:
9743:
9742:
9740:
9736:
9726:
9723:
9721:
9718:
9716:
9713:
9711:
9708:
9706:
9703:
9701:
9698:
9696:
9693:
9691:
9688:
9686:
9683:
9681:
9678:
9676:
9673:
9671:
9668:
9666:
9663:
9661:
9658:
9656:
9653:
9652:
9650:
9646:
9640:
9637:
9635:
9632:
9629:
9626:
9624:
9621:
9619:
9616:
9614:
9611:
9609:
9606:
9604:
9601:
9599:
9596:
9594:
9591:
9589:
9586:
9585:
9583:
9579:
9573:
9570:
9568:
9565:
9563:
9560:
9558:
9555:
9554:
9552:
9548:
9545:
9541:
9531:
9528:
9524:
9521:
9520:
9519:
9516:
9513:
9510:
9509:
9507:
9503:
9497:
9494:
9492:
9489:
9487:
9484:
9482:
9479:
9477:
9474:
9472:
9469:
9467:
9464:
9462:
9459:
9457:
9454:
9452:
9449:
9448:
9446:
9442:
9436:
9433:
9431:
9428:
9426:
9423:
9421:
9418:
9416:
9413:
9411:
9408:
9406:
9403:
9401:
9398:
9396:
9393:
9391:
9388:
9387:
9385:
9381:
9378:
9374:
9370:
9363:
9358:
9356:
9351:
9349:
9344:
9343:
9340:
9333:
9330:
9328:
9325:
9324:
9315:
9313:0-521-82650-0
9309:
9305:
9300:
9297:
9291:
9287:
9282:
9279:
9277:0-471-07411-X
9273:
9269:
9264:
9263:
9256:
9255:
9248:
9241:
9240:
9233:
9231:
9229:
9220:
9216:
9212:
9206:
9202:
9198:. This means
9195:
9191:
9187:
9181:
9174:
9170:
9166:
9160:
9156:
9152:
9147:
9141:
9132:
9128:
9118:
9115:
9113:
9110:
9108:
9105:
9103:
9100:
9098:
9095:
9094:
9088:
9086:
9082:
9081:singularities
9078:
9074:
9070:
9066:
9062:
9057:
9055:
9051:
9047:
9043:
9042:Taylor series
9039:
9035:
9030:
9027:
9025:
9021:
9017:
9013:
9008:
9006:
9002:
8998:
8994:
8990:
8986:
8982:
8978:
8974:
8970:
8966:
8962:
8958:
8953:
8951:
8947:
8941:
8931:
8913:
8910:
8907:
8903:
8899:
8896:
8893:
8888:
8884:
8861:
8858:
8855:
8849:
8843:
8838:
8834:
8830:
8827:
8824:
8818:
8807:
8804:
8801:
8794:
8788:
8785:
8782:
8778:
8772:
8769:
8766:
8762:
8758:
8752:
8741:
8734:
8728:
8724:
8715:
8706:
8704:
8699:
8695:
8693:
8689:
8685:
8680:
8678:
8674:
8670:
8666:
8662:
8658:
8648:
8635:
8632:
8629:
8622:
8608:
8600:
8597:
8593:
8587:
8580:
8576:
8571:
8564:
8558:
8555:
8535:
8531:
8522:
8519:
8515:
8508:
8502:
8499:
8493:
8465:
8460:
8450:
8442:
8438:
8420:
8418:
8373:
8370:
8367:
8360:
8346:
8338:
8335:
8331:
8327:
8321:
8315:
8312:
8294:
8288:
8285:
8279:
8258:
8256:
8252:
8236:
8230:
8224:
8221:
8218:
8215:
8209:
8203:
8200:
8194:
8191:
8187:
8173:
8150:
8147:
8144:
8141:
8138:
8135:
8125:
8111:
8106:
8085:
8079:
8070:of functions
8069:
8068:square matrix
8062:of dimension
8061:
8045:
8037:
8034:
8030:
8015:
8012:
7999:
7991:
7983:
7980:
7976:
7960:
7934:
7931:
7928:
7924:
7897:
7893:
7866:
7861:
7857:
7850:
7842:
7839:
7836:
7832:
7828:
7825:
7822:
7817:
7813:
7806:
7798:
7795:
7792:
7788:
7784:
7778:
7770:
7766:
7762:
7760:
7752:
7745:
7741:
7737:
7729:
7725:
7715:
7711:
7704:
7696:
7693:
7690:
7686:
7682:
7679:
7676:
7671:
7667:
7660:
7652:
7649:
7646:
7642:
7638:
7632:
7624:
7620:
7616:
7614:
7606:
7599:
7595:
7591:
7578:
7565:
7561:
7557:
7538:
7533:
7530:
7527:
7523:
7519:
7515:
7511:
7507:
7480:
7476:
7472:
7468:
7465:
7438:
7434:
7430:
7427:
7424:
7419:
7415:
7385:
7378:
7374:
7371:
7368:
7364:
7361:
7357:
7353:
7350:
7337:
7332:
7326:
7316:
7303:
7298:
7294:
7291:
7288:
7282:
7277:
7273:
7269:
7263:
7257:
7237:
7234:
7231:
7225:
7219:
7199:
7196:
7192:
7188:
7185:
7180:
7176:
7172:
7166:
7160:
7140:
7137:
7134:
7129:
7125:
7121:
7118:
7115:
7096:
7091:
7087:
7083:
7080:
7076:
7069:
7066:
7043:
7038:
7034:
7030:
7027:
7024:
7021:
7017:
7014:
7010:
7001:
6988:
6983:
6980:
6975:
6972:
6952:
6947:
6944:
6939:
6936:
6931:
6927:
6924:
6901:
6898:
6893:
6886:
6880:
6874:
6868:
6861:
6858:
6837:
6834:
6831:
6828:
6823:
6816:
6810:
6804:
6798:
6791:
6788:
6773:
6747:
6744:
6741:
6736:
6733:
6729:
6725:
6722:
6717:
6713:
6709:
6704:
6700:
6696:
6693:
6690:
6670:
6665:
6662:
6658:
6654:
6651:
6647:
6641:
6638:
6634:
6630:
6626:
6619:
6616:
6612:
6602:
6584:
6580:
6575:
6572:
6568:
6564:
6556:
6553:
6549:
6543:
6538:
6535:
6531:
6527:
6524:
6502:
6497:
6494:
6490:
6486:
6483:
6478:
6475:
6471:
6467:
6464:
6461:
6456:
6453:
6449:
6444:
6441:
6431:
6427:
6422:
6419:
6415:
6398:
6393:
6389:
6385:
6382:
6379:
6367:
6350:
6347:
6343:
6340:
6336:
6333:
6325:
6305:
6302:
6299:
6296:
6293:
6290:
6287:
6284:
6280:
6277:
6274:
6269:
6265:
6262:
6249:
6245:
6239:
6226:
6220:
6214:
6211:
6205:
6199:
6193:
6187:
6184:
6178:
6171:
6168:
6157:
6153:
6142:
6138:
6134:
6130:
6126:
6122:
6115:
6109:
6105:
6098:
6094:
6087:
6082:
6078:
6070:
6059:
6054:
6046:
6032:
6019:
6011:
6008:
6005:
5997:
5993:
5988:
5984:
5980:
5976:
5973:
5970:
5962:
5959:
5956:
5948:
5944:
5939:
5935:
5931:
5927:
5924:
5914:
5910:
5903:
5893:
5880:
5872:
5869:
5866:
5858:
5854:
5849:
5845:
5841:
5837:
5834:
5831:
5823:
5820:
5817:
5809:
5805:
5800:
5796:
5792:
5788:
5780:
5777:
5774:
5766:
5762:
5757:
5753:
5749:
5745:
5737:
5729:
5725:
5719:
5715:
5711:
5708:
5705:
5697:
5689:
5685:
5679:
5675:
5671:
5663:
5656:
5645:
5641:
5619:
5611:
5607:
5601:
5597:
5593:
5590:
5587:
5579:
5571:
5567:
5561:
5557:
5553:
5545:
5538:
5529:
5525:
5505:
5497:
5494:
5491:
5483:
5479:
5474:
5470:
5466:
5462:
5459:
5456:
5448:
5445:
5442:
5434:
5430:
5425:
5421:
5417:
5413:
5405:
5402:
5399:
5391:
5387:
5382:
5378:
5374:
5370:
5368:
5363:
5356:
5352:
5343:
5339:
5335:
5330:
5326:
5322:
5318:
5315:
5312:
5308:
5304:
5300:
5295:
5291:
5287:
5283:
5279:
5275:
5271:
5266:
5262:
5258:
5254:
5252:
5247:
5238:
5234:
5229:
5225:
5221:
5217:
5214:
5211:
5206:
5202:
5197:
5193:
5189:
5185:
5180:
5176:
5171:
5167:
5163:
5159:
5157:
5152:
5134:
5130:
5123:
5116:
5112:
5105:
5097:
5093:
5086:
5068:
5063:
5059:
5053:
5049:
5045:
5042:
5039:
5034:
5030:
5024:
5020:
5016:
5013:
4993:
4990:
4987:
4982:
4978:
4974:
4970:
4967:
4961:
4958:
4955:
4951:
4947:
4944:
4941:
4933:
4930:
4927:
4920:
4914:
4910:
4906:
4898:
4891:
4881:
4879:
4874:
4872:
4865:applies when
4864:
4860:
4846:
4842:
4835:
4831:
4825:
4822:
4813:
4800:
4796:
4776:
4772:
4765:
4748:
4742:
4736:
4733:
4727:
4721:
4716:
4712:
4708:
4702:
4695:
4692:
4686:
4683:
4680:
4676:
4672:
4669:
4666:
4660:
4649:
4646:
4643:
4636:
4630:
4626:
4622:
4616:
4605:
4598:
4579:
4573:
4565:
4556:
4547:
4538:
4525:
4521:
4512:
4508:
4501:
4497:
4477:
4468:
4465:
4459:
4456:
4451:
4447:
4443:
4437:
4434:
4428:
4425:
4420:
4416:
4407:
4404:
4400:
4391:
4375:
4370:
4364:
4361:
4358:
4355:
4348:
4342:
4338:
4334:
4329:
4323:
4320:
4317:
4314:
4307:
4301:
4297:
4287:
4283:
4278:
4272:
4267:
4253:
4248:
4244:
4240:
4237:
4234:
4230:
4223:
4218:
4214:
4210:
4205:
4201:
4187:
4179:
4174:
4160:
4155:
4152:
4148:
4142:
4138:
4134:
4129:
4126:
4122:
4116:
4112:
4093:
4088:
4087:
4086:
4080:
4071:
4065:
4061:
4057:
4052:
4039:
4026:
4023:
4020:
4017:
4014:
4011:
4006:
4002:
3981:
3978:
3975:
3972:
3969:
3966:
3962:
3959:
3955:
3952:
3948:
3945:
3930:
3913:
3910:
3904:
3901:
3896:
3893:
3889:
3883:
3879:
3855:
3852:
3846:
3843:
3838:
3835:
3831:
3825:
3821:
3798:
3792:
3789:
3786:
3783:
3776:
3770:
3766:
3743:
3737:
3734:
3731:
3728:
3721:
3715:
3711:
3702:
3697:
3693:
3687:
3683:
3678:
3673:
3671:
3666:
3651:
3648:
3642:
3639:
3635:
3622:
3616:
3603:
3598:
3595:
3591:
3585:
3582:
3579:
3575:
3571:
3568:
3564:
3558:
3555:
3551:
3545:
3541:
3536:
3531:
3527:
3524:
3518:
3515:
3511:
3505:
3496:
3474:
3471:
3465:
3462:
3458:
3441:
3437:
3433:
3429:
3415:
3411:
3382:
3377:
3374:
3370:
3364:
3360:
3351:
3347:
3338:
3325:
3320:
3316:
3312:
3308:
3303:
3299:
3294:
3291:
3288:
3285:
3281:
3278:
3275:
3272:
3252:
3247:
3243:
3239:
3235:
3230:
3226:
3221:
3216:
3213:
3210:
3206:
3201:
3196:
3193:
3189:
3175:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3130:
3126:
3122:
3119:
3114:
3110:
3106:
3103:
3098:
3094:
3073:
3070:
3067:
3064:
3060:
3057:
3053:
3050:
3046:
3043:
3039:
3036:
3032:
3029:
3025:
3022:
3018:
3015:
3006:
3005:
3001:
3000:
2997:
2995:
2991:
2985:
2981:
2976:
2972:
2964:
2959:
2946:
2943:
2938:
2934:
2928:
2924:
2920:
2917:
2914:
2909:
2905:
2899:
2895:
2891:
2888:
2883:
2879:
2875:
2870:
2866:
2858:
2840:
2836:
2830:
2826:
2822:
2819:
2816:
2811:
2807:
2801:
2797:
2793:
2790:
2785:
2781:
2777:
2772:
2768:
2760:
2751:
2734:
2731:
2726:
2723:
2719:
2713:
2709:
2703:
2699:
2695:
2692:
2689:
2684:
2681:
2677:
2671:
2667:
2661:
2657:
2653:
2648:
2645:
2641:
2637:
2632:
2628:
2624:
2619:
2616:
2612:
2606:
2602:
2588:
2582:
2578:
2574:
2567:
2550:
2547:
2539:
2532:
2526:
2522:
2518:
2515:
2512:
2508:
2505:
2499:
2495:
2491:
2487:
2484:
2478:
2474:
2470:
2467:
2462:
2458:
2448:
2445:
2442:
2436:
2425:
2419:
2415:
2409:
2405:
2401:
2396:
2392:
2388:
2381:
2364:
2361:
2353:
2346:
2340:
2336:
2332:
2329:
2326:
2322:
2319:
2313:
2309:
2305:
2301:
2298:
2292:
2288:
2284:
2281:
2276:
2272:
2263:
2253:
2242:
2236:
2231:
2227:
2210:
2206:
2199:
2195:
2184:
2180:
2173:
2156:
2150:
2142:
2138:
2132:
2128:
2124:
2121:
2118:
2112:
2104:
2100:
2094:
2090:
2086:
2080:
2072:
2068:
2057:
2053:
2049:
2045:
2032:
2023:
2019:
2014:
2010:
2006:
2001:
1998:
1994:
1987:
1983:
1979:
1975:
1965:
1952:
1946:
1940:
1937:
1934:
1931:
1908:
1902:
1899:
1891:
1884:
1877:
1869:
1865:
1861:
1858:
1855:
1851:
1848:
1841:
1833:
1829:
1825:
1821:
1818:
1811:
1803:
1799:
1795:
1792:
1786:
1778:
1774:
1753:
1745:
1741:
1737:
1731:
1727:
1718:
1710:
1706:
1702:
1699:
1696:
1690:
1687:
1683:
1675:
1667:
1663:
1659:
1653:
1645:
1641:
1637:
1634:
1625:
1623:
1619:
1615:
1611:
1607:
1602:
1600:
1596:
1590:
1586:
1580:
1562:
1560:
1559:zero function
1554:
1549:
1545:
1540:
1530:
1525:
1521:
1517:
1510:
1493:
1485:
1481:
1477:
1471:
1467:
1458:
1450:
1446:
1442:
1439:
1436:
1430:
1427:
1423:
1415:
1407:
1403:
1399:
1393:
1385:
1381:
1372:
1368:
1364:
1360:
1355:
1330:
1326:
1320:
1316:
1309:
1302:
1298:
1292:
1288:
1276:
1272:
1268:
1265:
1262:
1257:
1253:
1237:
1216:
1212:
1208:
1202:
1198:
1183:
1179:
1178:th derivative
1172:
1164:
1158:
1148:
1146:
1142:
1133:
1131:
1127:
1123:
1115:
1112:, as it is a
1111:
1110:
1105:
1104:zero function
1101:
1097:
1096:constant term
1091:
1087:
1082:
1078:
1068:
1066:
1062:
1058:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
997:
995:
991:
987:
977:
975:
971:
967:
963:
958:
947:
943:
938:
932:
928:
920:
915:
911:
904:
897:
876:
870:
867:
859:
852:
845:
837:
833:
829:
826:
822:
819:
812:
804:
800:
796:
792:
789:
782:
774:
770:
766:
763:
757:
749:
745:
737:of the form
736:
732:
728:
724:
720:
708:
703:
701:
696:
694:
689:
688:
686:
685:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
598:
591:
590:
586:
585:
578:
575:
573:
570:
568:
565:
564:
560:
557:
555:
552:
550:
547:
545:
542:
540:
537:
533:
530:
529:
528:
525:
523:
522:Finite volume
520:
516:
513:
512:
511:
508:
504:
498:
495:
493:
490:
488:
484:
482:
479:
476:
475:
468:
467:
459:
456:
454:
451:
447:
443:
441:
438:
436:
432:
428:
425:
423:
420:
418:
415:
411:
408:
406:
403:
401:
398:
396:
393:
392:
391:
388:
386:
383:
382:
375:
374:
367:
364:
362:
359:
357:
354:
352:
349:
348:
342:
341:
337:
336:
328:
325:
321:
318:
317:
316:
313:
312:
308:
302:
301:
295:
294:
283:
280:
279:
275:
272:
270:
267:
266:
264:
263:
259:
258:
252:
248:
245:
243:
240:
237:
235:
232:
231:
227:
224:
223:
222:
221:
217:
216:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
176:
174:
173:
165:
164:
160:
159:
154:
144:
141:
139:
136:
135:
134:
133:
130:
127:
126:
121:
118:
116:
113:
111:
108:
107:
106:
105:
102:
99:
98:
93:
90:
88:
84:
82:
79:
77:
74:
72:
69:
68:
67:
66:
60:
57:
55:
52:
51:
49:
48:
40:
39:
35:
34:
31:
28:
27:
22:
9841:Martin Kutta
9796:Émile Picard
9776:Isaac Newton
9690:Euler method
9660:Substitution
9424:
9303:
9285:
9267:
9252:
9247:
9237:
9218:
9214:
9210:
9204:
9200:
9193:
9189:
9185:
9179:
9172:
9168:
9164:
9158:
9154:
9150:
9140:
9131:
9058:
9053:
9046:power series
9033:
9031:
9028:
9009:
8954:
8949:
8943:
8712:
8700:
8696:
8681:
8669:Émile Picard
8654:
8421:
8259:
8109:
8060:vector space
8016:
8013:
7958:
7566:
7559:
7555:
7338:
7334:
7002:
6779:
6601:product rule
6429:
6423:
6417:
6413:
6247:
6243:
6240:
6155:
6151:
6148:
6139:
6132:
6128:
6124:
6120:
6113:
6107:
6103:
6096:
6092:
6085:
6068:
6057:
6052:
6044:
6033:
5912:
5908:
5901:
5894:
5643:
5639:
5524:product rule
5132:
5128:
5121:
5114:
5110:
5103:
5095:
5091:
5084:
4882:
4875:
4844:
4840:
4833:
4829:
4823:
4820:
4801:
4794:
4774:
4770:
4763:
4585:
4563:
4554:
4545:
4536:
4523:
4519:
4510:
4506:
4499:
4495:
4492:
4285:
4281:
4270:
4185:
4177:
4091:
4078:
4069:
4063:
4059:
4055:
4040:
3936:
3695:
3691:
3685:
3681:
3674:
3667:
3620:
3617:
3439:
3435:
3431:
3427:
3413:
3409:
3346:simple roots
3343:
3173:
3007:
2994:vector space
2983:
2979:
2960:
2758:
2749:
2586:
2583:
2576:
2572:
2565:
2449:
2443:
2440:
2434:
2423:
2417:
2413:
2407:
2397:
2390:
2386:
2379:
2261:
2259:
2240:
2239:| >
2234:
2229:
2225:
2208:
2204:
2197:
2193:
2182:
2178:
2171:
2055:
2051:
2047:
2043:
2024:
2017:
2013:vector space
2004:
2002:
1996:
1992:
1985:
1981:
1977:
1973:
1966:
1626:
1610:real numbers
1606:vector space
1603:
1588:
1584:
1578:
1563:
1552:
1547:
1543:
1538:
1528:
1523:
1519:
1515:
1508:
1366:
1365:or, simply,
1362:
1358:
1356:
1162:
1160:
1145:vector space
1136:
1134:
1125:
1117:
1107:
1095:
1089:
1085:
1080:
1075:The highest
1074:
998:
983:
965:
959:
945:
941:
930:
926:
918:
913:
909:
902:
895:
722:
716:
666:Émile Picard
651:Martin Kutta
641:George Green
601:Isaac Newton
433: /
429: /
249: /
203:
115:Chaos theory
9598:Phase space
9456:Homogeneous
8995:, and many
8957:polynomials
8172:exponential
8105:determinant
6101:, and then
4504:satisfying
3668:As, by the
1618:free module
1557:is not the
1109:homogeneous
1013:integration
719:mathematics
559:Runge–Kutta
304:Difference
247:Homogeneous
59:Engineering
9826:John Crank
9655:Inspection
9518:Stochastic
9512:Difference
9486:Autonomous
9430:Non-linear
9420:Fractional
9383:Operations
9208:, so that
9123:References
9069:indefinite
9065:derivative
9054:vice versa
9024:algorithms
9016:derivative
8657:quadrature
8170:equal the
7558:= 1, ...,
7329:See also:
6426:reciprocal
5642:= 1, ...,
2982:= 0, ...,
2965:, one has
2594:such that
2421:such that
2244:for every
2221:such that
1236:univariate
1001:polynomial
986:quadrature
676:John Crank
477:Inspection
431:Asymptotic
315:Stochastic
234:Autonomous
209:Non-linear
199:Fractional
9630:solutions
9588:Wronskian
9543:Solutions
9471:Decoupled
9435:Holonomic
9183:, namely
9020:integrals
8965:logarithm
8911:−
8897:…
8828:⋯
8805:−
8786:−
8770:−
8661:integrals
8598:−
8572:∫
8520:−
8336:−
8328:∫
8225:
8204:
8142:∫
7826:⋯
7730:⋮
7680:⋯
7428:…
7372:…
7292:−
7289:α
7235:α
7056:That is
6965:that is
6940:−
6734:−
6723:∫
6663:−
6639:−
6573:−
6536:−
6525:−
6495:−
6476:−
6462:−
6454:−
6341:∫
6288:
6009:−
5974:⋯
5960:−
5870:−
5835:⋯
5821:−
5778:−
5709:⋯
5591:⋯
5528:induction
5495:−
5460:⋯
5446:−
5403:−
5357:⋮
5316:⋯
5215:⋯
5043:⋯
4959:−
4945:⋯
4931:−
4684:−
4670:⋯
4647:−
4466:β
4460:
4435:β
4429:
4405:α
4362:β
4359:−
4356:α
4321:β
4315:α
4235:−
4153:β
4127:α
3905:
3847:
3787:−
3652:α
3649:−
3596:α
3583:−
3556:α
3528:α
3525:−
3475:α
3472:−
3375:α
3289:
3276:
3211:−
3136:−
3104:−
3051:−
3023:−
2918:⋯
2820:⋯
2724:α
2710:α
2693:⋯
2682:α
2668:α
2646:α
2638:α
2617:α
2516:⋯
2330:⋯
2122:⋯
1859:⋯
1700:⋯
1620:over the
1608:over the
1440:⋯
1313:∂
1310:⋯
1285:∂
1266:⋯
1249:∂
1184:of order
1165:of order
1025:logarithm
990:integrals
964:(ODE). A
827:⋯
417:Wronskian
395:Dirichlet
138:Economics
81:Chemistry
71:Astronomy
9861:Category
9738:Examples
9628:Integral
9400:Ordinary
9091:See also
9061:calculus
8999:such as
8103:, whose
8031:′
7977:′
7914:and the
7746:′
7600:′
7516:′
7469:′
7365:″
7354:′
7077:′
7018:′
6928:′
6862:′
6792:′
6445:′
6266:′
6172:′
5989:′
5940:′
5850:′
5801:′
5758:′
5475:′
5426:′
5383:′
5344:′
5331:′
5309:′
5296:′
5280:′
5267:′
5230:′
5198:′
5172:′
4971:′
4849:, where
4696:′
3963:′
3949:″
3061:′
3047:″
3033:‴
3019:⁗
3002:Example
2963:distinct
2509:″
2488:′
2323:″
2302:′
1852:″
1822:′
1518:), ...,
1367:operator
1139:solution
1128:if only
1053:calculus
1019:such as
944:′, ...,
823:″
793:′
735:equation
527:Galerkin
427:Lyapunov
338:Solution
282:Notation
274:Operator
260:Features
179:Ordinary
9466:Coupled
9405:Partial
8928:
8876:
8164:
8127:
8101:
8072:
7949:
7916:
7912:
7885:
7551:
7499:
7495:
7457:
7453:
7407:
7403:
7342:
6776:Example
6597:
6517:
6364:is any
6160:, is:
6091:, ...,
6051:, ...,
5907:, ...,
5127:, ...,
5109:, ...,
5090:, ...,
4769:, ...,
4522:′(0) =
2992:of the
2571:, ...,
2426:(0) = 1
2385:, ...,
2203:, ...,
2177:, ...,
1612:or the
1537:is the
1369:) is a
1173:to its
907:, ...,
400:Neumann
184:Partial
92:Geology
87:Biology
76:Physics
9481:Degree
9425:Linear
9310:
9292:
9274:
9077:limits
8977:cosine
8874:where
7883:where
6914:gives
6760:where
6411:where
6318:where
6119:+ ⋯ +
6083:gives
6066:, the
5648:, and
5081:where
4838:, and
4787:, and
4761:where
4509:(0) =
4279:roots
4273:< 0
4100:, and
4094:> 0
3407:, and
3395:where
3177:, and
2377:where
2223:|
2169:where
2009:kernel
2005:kernel
1599:scalar
1506:where
1061:limits
1033:cosine
892:where
587:People
499:
446:Series
204:Linear
43:Fields
9530:Delay
9476:Order
9203:′ = −
8114:, or
6250:) = 0
3412:<
2990:basis
2450:Let
1539:order
1081:order
725:is a
487:Euler
405:Robin
327:Delay
269:Order
242:Exact
168:Types
36:Scope
9308:ISBN
9290:ISBN
9272:ISBN
9188:′ −
9153:′ −
9071:and
9018:and
9003:and
8991:and
8973:sine
8671:and
8017:Let
7553:for
7497:and
7153:and
6599:the
6326:and
5637:for
5526:and
5006:is
4843:sin(
4832:cos(
4561:and
4543:and
4517:and
4076:and
4051:real
4049:are
4045:and
3871:and
3758:and
2416:′ =
2050:) =
2003:The
1980:) =
1622:ring
1564:Let
1047:and
1029:sine
924:and
721:, a
594:List
9192:= (
9190:hfy
9056:.
8222:exp
8201:exp
8174:of
8112:= 1
7963:")
7562:– 1
6515:As
6368:of
6285:log
5646:– 1
4457:sin
4426:cos
4392:as
4268:If
4180:= 0
4175:If
4089:If
4062:− 4
4041:If
3929:.
3902:sin
3844:cos
3813:by
3623:+ 1
3286:sin
3273:cos
2986:– 1
2438:is
2248:in
2020:= 0
1990:or
1582:or
1561:).
717:In
9863::
9227:^
9217:=
9213:=
9196:)′
9194:hy
9186:hy
9180:hy
9167:=
9157:=
9155:fy
9083:,
9067:,
9032:A
9007:.
8987:,
8983:,
8979:,
8975:,
8971:,
8967:,
8963:,
8959:,
8944:A
8705:.
8694:.
8679:.
8419:.
8257:.
7564:.
6416:=
6154:′(
6137:.
6106:=
5530:)
4873:.
4845:ax
4834:ax
4827:,
4286:βi
4284:±
4188:/2
4085:.
4058:=
3696:ib
3694:–
3686:ib
3684:+
3497:,
3438:−
3434:)(
3170:,
3154:0.
2947:0.
2735:0.
2252:.
2196:,
2044:Ly
2029:,
2022:.
2018:Ly
2000:.
1995:=
1993:Ly
1974:Ly
1585:Lf
1579:Lf
1357:A
1161:A
1135:A
1063:,
1059:,
1043:,
1039:,
1035:,
1031:,
1027:,
1023:,
1011:,
976:.
957:.
9361:e
9354:t
9347:v
9219:e
9215:e
9211:h
9205:f
9201:h
9175:)
9173:x
9171:(
9169:h
9165:h
9159:g
9151:y
8914:1
8908:n
8904:a
8900:,
8894:,
8889:0
8885:a
8862:,
8859:0
8856:=
8853:)
8850:x
8847:(
8844:y
8839:0
8835:a
8831:+
8825:+
8822:)
8819:x
8816:(
8811:)
8808:1
8802:n
8799:(
8795:y
8789:1
8783:n
8779:x
8773:1
8767:n
8763:a
8759:+
8756:)
8753:x
8750:(
8745:)
8742:n
8739:(
8735:y
8729:n
8725:x
8636:.
8633:t
8630:d
8626:)
8623:t
8620:(
8616:b
8612:)
8609:t
8606:(
8601:1
8594:U
8588:x
8581:0
8577:x
8568:)
8565:x
8562:(
8559:U
8556:+
8550:0
8546:y
8541:)
8536:0
8532:x
8528:(
8523:1
8516:U
8512:)
8509:x
8506:(
8503:U
8500:=
8497:)
8494:x
8491:(
8487:y
8466:,
8461:0
8456:y
8451:=
8448:)
8443:0
8439:x
8435:(
8431:y
8400:0
8396:y
8374:,
8371:x
8368:d
8364:)
8361:x
8358:(
8354:b
8350:)
8347:x
8344:(
8339:1
8332:U
8325:)
8322:x
8319:(
8316:U
8313:+
8307:0
8303:y
8298:)
8295:x
8292:(
8289:U
8286:=
8283:)
8280:x
8277:(
8273:y
8262:U
8237:.
8234:)
8231:B
8228:(
8219:A
8216:=
8213:)
8210:B
8207:(
8195:x
8192:d
8188:d
8176:B
8168:U
8151:x
8148:d
8145:A
8139:=
8136:B
8120:A
8116:A
8110:n
8089:)
8086:x
8083:(
8080:U
8064:n
8046:.
8042:u
8038:A
8035:=
8027:u
8000:.
7996:b
7992:+
7988:y
7984:A
7981:=
7973:y
7961:)
7959:x
7957:(
7953:x
7935:j
7932:,
7929:i
7925:a
7898:n
7894:b
7867:,
7862:n
7858:y
7854:)
7851:x
7848:(
7843:n
7840:,
7837:n
7833:a
7829:+
7823:+
7818:1
7814:y
7810:)
7807:x
7804:(
7799:1
7796:,
7793:n
7789:a
7785:+
7782:)
7779:x
7776:(
7771:n
7767:b
7763:=
7756:)
7753:x
7750:(
7742:n
7738:y
7716:n
7712:y
7708:)
7705:x
7702:(
7697:n
7694:,
7691:1
7687:a
7683:+
7677:+
7672:1
7668:y
7664:)
7661:x
7658:(
7653:1
7650:,
7647:1
7643:a
7639:+
7636:)
7633:x
7630:(
7625:1
7621:b
7617:=
7610:)
7607:x
7604:(
7596:1
7592:y
7573:n
7569:n
7560:k
7556:i
7539:,
7534:1
7531:+
7528:i
7524:y
7520:=
7512:i
7508:y
7481:1
7477:y
7473:=
7466:y
7439:k
7435:y
7431:,
7425:,
7420:1
7416:y
7389:)
7386:k
7383:(
7379:y
7375:,
7369:,
7362:y
7358:,
7351:y
7304:.
7299:x
7295:1
7283:+
7278:2
7274:x
7270:=
7267:)
7264:x
7261:(
7258:y
7238:,
7232:=
7229:)
7226:1
7223:(
7220:y
7200:.
7197:x
7193:/
7189:c
7186:+
7181:2
7177:x
7173:=
7170:)
7167:x
7164:(
7161:y
7141:,
7138:c
7135:+
7130:3
7126:x
7122:=
7119:y
7116:x
7097:,
7092:2
7088:x
7084:3
7081:=
7074:)
7070:y
7067:x
7064:(
7044:.
7039:2
7035:x
7031:3
7028:=
7025:y
7022:+
7015:y
7011:x
6989:.
6984:x
6981:c
6976:=
6973:y
6953:,
6948:x
6945:1
6937:=
6932:y
6925:y
6902:0
6899:=
6894:x
6890:)
6887:x
6884:(
6881:y
6875:+
6872:)
6869:x
6866:(
6859:y
6838:.
6835:x
6832:3
6829:=
6824:x
6820:)
6817:x
6814:(
6811:y
6805:+
6802:)
6799:x
6796:(
6789:y
6770:f
6766:F
6762:c
6748:,
6745:x
6742:d
6737:F
6730:e
6726:g
6718:F
6714:e
6710:+
6705:F
6701:e
6697:c
6694:=
6691:y
6671:.
6666:F
6659:e
6655:g
6652:=
6648:)
6642:F
6635:e
6631:y
6627:(
6620:x
6617:d
6613:d
6585:,
6581:)
6576:F
6569:e
6565:(
6557:x
6554:d
6550:d
6544:=
6539:F
6532:e
6528:f
6503:.
6498:F
6491:e
6487:g
6484:=
6479:F
6472:e
6468:f
6465:y
6457:F
6450:e
6442:y
6430:e
6418:e
6414:c
6399:,
6394:F
6390:e
6386:c
6383:=
6380:y
6370:f
6351:x
6348:d
6344:f
6337:=
6334:F
6320:k
6306:,
6303:F
6300:+
6297:k
6294:=
6291:y
6281:,
6278:f
6275:=
6270:y
6263:y
6248:x
6246:(
6244:g
6227:.
6224:)
6221:x
6218:(
6215:g
6212:+
6209:)
6206:x
6203:(
6200:y
6197:)
6194:x
6191:(
6188:f
6185:=
6182:)
6179:x
6176:(
6169:y
6158:)
6156:x
6152:y
6133:n
6129:y
6125:n
6121:u
6117:1
6114:y
6111:1
6108:u
6104:y
6097:n
6093:u
6089:1
6086:u
6072:i
6069:y
6064:f
6058:n
6055:′
6053:u
6049:1
6047:′
6045:u
6040:n
6036:0
6020:.
6015:)
6012:1
6006:n
6003:(
5998:n
5994:y
5985:n
5981:u
5977:+
5971:+
5966:)
5963:1
5957:n
5954:(
5949:1
5945:y
5936:1
5932:u
5928:=
5925:f
5913:n
5909:y
5905:1
5902:y
5897:y
5881:.
5876:)
5873:1
5867:n
5864:(
5859:n
5855:y
5846:n
5842:u
5838:+
5832:+
5827:)
5824:1
5818:n
5815:(
5810:2
5806:y
5797:2
5793:u
5789:+
5784:)
5781:1
5775:n
5772:(
5767:1
5763:y
5754:1
5750:u
5746:+
5741:)
5738:n
5735:(
5730:n
5726:y
5720:n
5716:u
5712:+
5706:+
5701:)
5698:n
5695:(
5690:1
5686:y
5680:1
5676:u
5672:=
5667:)
5664:n
5661:(
5657:y
5644:n
5640:i
5623:)
5620:i
5617:(
5612:n
5608:y
5602:n
5598:u
5594:+
5588:+
5583:)
5580:i
5577:(
5572:1
5568:y
5562:1
5558:u
5554:=
5549:)
5546:i
5543:(
5539:y
5506:,
5501:)
5498:2
5492:n
5489:(
5484:n
5480:y
5471:n
5467:u
5463:+
5457:+
5452:)
5449:2
5443:n
5440:(
5435:2
5431:y
5422:2
5418:u
5414:+
5409:)
5406:2
5400:n
5397:(
5392:1
5388:y
5379:1
5375:u
5371:=
5364:0
5340:n
5336:y
5327:n
5323:u
5319:+
5313:+
5305:2
5301:y
5292:2
5288:u
5284:+
5276:1
5272:y
5263:1
5259:u
5255:=
5248:0
5239:n
5235:y
5226:n
5222:u
5218:+
5212:+
5207:2
5203:y
5194:2
5190:u
5186:+
5181:1
5177:y
5168:1
5164:u
5160:=
5153:0
5139:y
5133:n
5129:u
5125:1
5122:u
5115:n
5111:u
5107:1
5104:u
5099:)
5096:n
5092:y
5088:1
5085:y
5083:(
5069:,
5064:n
5060:y
5054:n
5050:u
5046:+
5040:+
5035:1
5031:y
5025:1
5021:u
5017:=
5014:y
4994:0
4991:=
4988:y
4983:n
4979:a
4975:+
4968:y
4962:1
4956:n
4952:a
4948:+
4942:+
4937:)
4934:1
4928:n
4925:(
4921:y
4915:1
4911:a
4907:+
4902:)
4899:n
4896:(
4892:y
4867:f
4855:a
4851:n
4847:)
4841:x
4836:)
4830:x
4824:e
4821:x
4816:f
4808:f
4804:f
4797:)
4795:x
4793:(
4789:y
4785:x
4781:f
4775:n
4771:a
4767:1
4764:a
4749:,
4746:)
4743:x
4740:(
4737:f
4734:=
4731:)
4728:x
4725:(
4722:y
4717:n
4713:a
4709:+
4706:)
4703:x
4700:(
4693:y
4687:1
4681:n
4677:a
4673:+
4667:+
4664:)
4661:x
4658:(
4653:)
4650:1
4644:n
4641:(
4637:y
4631:1
4627:a
4623:+
4620:)
4617:x
4614:(
4609:)
4606:n
4603:(
4599:y
4588:n
4576:0
4567:2
4564:c
4558:1
4555:c
4549:2
4546:d
4540:1
4537:d
4532:0
4527:2
4524:d
4520:y
4514:1
4511:d
4507:y
4502:)
4500:x
4498:(
4496:y
4478:.
4475:)
4472:)
4469:x
4463:(
4452:2
4448:c
4444:+
4441:)
4438:x
4432:(
4421:1
4417:c
4413:(
4408:x
4401:e
4376:,
4371:x
4368:)
4365:i
4353:(
4349:e
4343:2
4339:c
4335:+
4330:x
4327:)
4324:i
4318:+
4312:(
4308:e
4302:1
4298:c
4282:α
4271:D
4254:.
4249:2
4245:/
4241:x
4238:a
4231:e
4227:)
4224:x
4219:2
4215:c
4211:+
4206:1
4202:c
4198:(
4186:a
4184:−
4178:D
4161:.
4156:x
4149:e
4143:2
4139:c
4135:+
4130:x
4123:e
4117:1
4113:c
4102:β
4098:α
4092:D
4082:2
4079:c
4073:1
4070:c
4064:b
4060:a
4056:D
4047:b
4043:a
4027:.
4024:b
4021:+
4018:r
4015:a
4012:+
4007:2
4003:r
3982:,
3979:0
3976:=
3973:y
3970:b
3967:+
3960:y
3956:a
3953:+
3946:y
3917:)
3914:x
3911:b
3908:(
3897:x
3894:a
3890:e
3884:k
3880:x
3859:)
3856:x
3853:b
3850:(
3839:x
3836:a
3832:e
3826:k
3822:x
3799:x
3796:)
3793:b
3790:i
3784:a
3781:(
3777:e
3771:k
3767:x
3744:x
3741:)
3738:b
3735:i
3732:+
3729:a
3726:(
3722:e
3716:k
3712:x
3692:a
3682:a
3664:.
3643:x
3640:d
3636:d
3621:k
3604:,
3599:x
3592:e
3586:1
3580:k
3576:x
3572:k
3569:=
3565:)
3559:x
3552:e
3546:k
3542:x
3537:(
3532:)
3519:x
3516:d
3512:d
3506:(
3491:P
3487:,
3466:x
3463:d
3459:d
3446:m
3442:)
3440:α
3436:t
3432:t
3430:(
3428:P
3423:m
3419:α
3414:m
3410:k
3405:m
3401:α
3397:k
3383:,
3378:x
3371:e
3365:k
3361:x
3326:.
3321:x
3317:e
3313:x
3309:,
3304:x
3300:e
3295:,
3292:x
3282:,
3279:x
3253:.
3248:x
3244:e
3240:x
3236:,
3231:x
3227:e
3222:,
3217:x
3214:i
3207:e
3202:,
3197:x
3194:i
3190:e
3179:1
3174:i
3172:−
3168:i
3151:=
3148:1
3145:+
3142:z
3139:2
3131:2
3127:z
3123:2
3120:+
3115:3
3111:z
3107:2
3099:4
3095:z
3074:0
3071:=
3068:y
3065:+
3058:y
3054:2
3044:y
3040:2
3037:+
3030:y
3026:2
3016:y
2984:n
2980:x
2967:n
2944:=
2939:n
2935:t
2929:n
2925:a
2921:+
2915:+
2910:2
2906:t
2900:2
2896:a
2892:+
2889:t
2884:1
2880:a
2876:+
2871:0
2867:a
2841:n
2837:t
2831:n
2827:a
2823:+
2817:+
2812:2
2808:t
2802:2
2798:a
2794:+
2791:t
2786:1
2782:a
2778:+
2773:0
2769:a
2755:α
2750:e
2732:=
2727:x
2720:e
2714:n
2704:n
2700:a
2696:+
2690:+
2685:x
2678:e
2672:2
2662:2
2658:a
2654:+
2649:x
2642:e
2633:1
2629:a
2625:+
2620:x
2613:e
2607:0
2603:a
2592:α
2587:e
2577:n
2573:a
2569:0
2566:a
2551:0
2548:=
2543:)
2540:n
2537:(
2533:y
2527:n
2523:a
2519:+
2513:+
2506:y
2500:2
2496:a
2492:+
2485:y
2479:1
2475:a
2471:+
2468:y
2463:0
2459:a
2444:e
2441:c
2435:e
2430:n
2424:f
2418:f
2414:f
2408:e
2391:n
2387:a
2383:1
2380:a
2365:0
2362:=
2357:)
2354:n
2351:(
2347:y
2341:n
2337:a
2333:+
2327:+
2320:y
2314:2
2310:a
2306:+
2299:y
2293:1
2289:a
2285:+
2282:y
2277:0
2273:a
2250:I
2246:x
2241:k
2237:)
2235:x
2233:(
2230:n
2226:a
2219:k
2215:I
2209:n
2205:a
2201:0
2198:a
2194:b
2189:I
2183:n
2179:c
2175:1
2172:c
2157:,
2154:)
2151:x
2148:(
2143:n
2139:S
2133:n
2129:c
2125:+
2119:+
2116:)
2113:x
2110:(
2105:1
2101:S
2095:1
2091:c
2087:+
2084:)
2081:x
2078:(
2073:0
2069:S
2058:)
2056:x
2054:(
2052:b
2048:x
2046:(
2039:n
2035:L
2027:n
1997:b
1988:)
1986:x
1984:(
1982:b
1978:x
1976:(
1969:y
1953:.
1950:)
1947:x
1944:(
1941:b
1938:=
1935:y
1932:L
1912:)
1909:x
1906:(
1903:b
1900:=
1895:)
1892:n
1889:(
1885:y
1881:)
1878:x
1875:(
1870:n
1866:a
1862:+
1856:+
1849:y
1845:)
1842:x
1839:(
1834:2
1830:a
1826:+
1819:y
1815:)
1812:x
1809:(
1804:1
1800:a
1796:+
1793:y
1790:)
1787:x
1784:(
1779:0
1775:a
1754:,
1746:n
1742:x
1738:d
1732:n
1728:d
1722:)
1719:x
1716:(
1711:n
1707:a
1703:+
1697:+
1691:x
1688:d
1684:d
1679:)
1676:x
1673:(
1668:1
1664:a
1660:+
1657:)
1654:x
1651:(
1646:0
1642:a
1638:=
1635:L
1591:)
1589:X
1587:(
1574:f
1570:L
1566:L
1555:)
1553:x
1551:(
1548:n
1544:a
1535:n
1531:)
1529:x
1527:(
1524:n
1520:a
1516:x
1514:(
1512:0
1509:a
1494:,
1486:n
1482:x
1478:d
1472:n
1468:d
1462:)
1459:x
1456:(
1451:n
1447:a
1443:+
1437:+
1431:x
1428:d
1424:d
1419:)
1416:x
1413:(
1408:1
1404:a
1400:+
1397:)
1394:x
1391:(
1386:0
1382:a
1352:n
1331:n
1327:i
1321:n
1317:x
1303:1
1299:i
1293:1
1289:x
1277:n
1273:i
1269:+
1263:+
1258:1
1254:i
1217:i
1213:x
1209:d
1203:i
1199:d
1186:i
1176:i
1167:i
1092:)
1090:x
1088:(
1086:b
955:x
951:y
946:y
942:y
933:)
931:x
929:(
927:b
921:)
919:x
917:(
914:n
910:a
905:)
903:x
901:(
899:0
896:a
880:)
877:x
874:(
871:b
868:=
863:)
860:n
857:(
853:y
849:)
846:x
843:(
838:n
834:a
830:+
820:y
816:)
813:x
810:(
805:2
801:a
797:+
790:y
786:)
783:x
780:(
775:1
771:a
767:+
764:y
761:)
758:x
755:(
750:0
746:a
706:e
699:t
692:v
505:)
501:(
23:.
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