5531:
7892:
5155:
5902:
7593:
2756:
8657:
8708:
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.
1359:
7346:
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.
1158:. 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.
3625:
5526:{\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}}}
3683:, 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.
6151:
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.
1933:
8395:
1775:
901:
7350:
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
3095:
1515:
4770:
8883:
5662:
2572:
2386:
6041:
4397:
1003:. 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,
5015:
2608:
4499:
7598:
5646:
8021:
6606:
2968:
2178:
6692:
8492:
2864:
9249:
6327:
8487:
7412:
6974:
1252:
5160:
4182:
7887:{\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}}}
7325:
3274:
8067:
3347:
8258:
6859:
5090:
6524:
3511:
6923:
6769:
4003:
3175:
1615:
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
6248:
4275:
6373:
1243:
8937:
8173:
7560:
3673:
3496:
7462:
3938:
3880:
8278:
7118:
7065:
3404:
1641:
7010:
8424:
3822:
3767:
1387:
7221:
9086:, 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,
8025:
The solving method is similar to that of a single first order linear differential equations, but with complications stemming from noncommutativity of matrix multiplication.
7504:
7259:
7162:
4048:
6420:
4604:
1974:
7958:
8730:
7921:
8110:
1780:
17:
1127:
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
751:
9370:
4303:
4406:
995:
A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by
5931:
5544:
4897:
1062:. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of
9765:
30:
This article is about linear differential equations with one independent variable. For similar equations with two or more independent variables, see
2464:
2278:
9770:
8966:
Most functions that are commonly considered in mathematics are holonomic or quotients of holonomic functions. In fact, holonomic functions include
3686:
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
2872:
2074:
715:
6617:
9633:
9107:
9063:
is zero). There are efficient algorithms for both conversions, that is for computing the recurrence relation from the differential equation, and
456:
2774:
9760:
8701:
of degree at least five cannot, in general, be solved by radicals. This analogy extends to the proof methods and motivates the denomination of
7977:
2406:
are (real or complex) numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.
31:
9070:
It follows that, if one represents (in a computer) holonomic functions by their defining differential equations and initial conditions, most
1384:
of basic differential operators, with differentiable functions as coefficients. In the univariate case, a linear operator has thus the form
8436:
6266:
9363:
4118:
1154:
of a differential equation is a function that satisfies the equation. The solutions of a homogeneous linear differential equation form a
7264:
3195:
2980:
distinct solutions that are not necessarily real, even if the coefficients of the equation are real. These solutions can be shown to be
9420:
8727:
are examples of equations of any order, with variable coefficients, that can be solved explicitly. These are the equations of the form
7587:
3279:
199:
8192:
6928:
5897:{\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)}.}
5020:
9533:
9037:
for computing the differential equation of the result of any of these operations, knowing the differential equations of the input.
6697:
6531:
330:
8031:
6794:
3100:
6447:
9356:
2751:{\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.}
9577:
4204:
3021:
2041:
1978:
There may be several variants to this notation; in particular the variable of differentiation may appear explicitly or not in
1604:, if one needs to specify the variable (this must not be confused with a multiplication). A linear differential operator is a
371:
9471:
9304:
5130:
are arbitrary constants. The method of variation of constants takes its name from the following idea. Instead of considering
1202:
261:
7586:
differential equations may normally be solved for the derivatives of the unknown functions. If it is not the case this is a
9755:
708:
279:
163:
9665:
6174:
9680:
9461:
4869:
582:
236:
3366:
6979:
2867:
9466:
9430:
9322:
9286:
7167:
1119:
257:
209:
9033:
of holonomic functions are holonomic. Moreover, these closure properties are effective, in the sense that there are
8652:{\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.}
9528:
9440:
9055:
at a point of a holonomic function form a holonomic sequence. Conversely, if the sequence of the coefficients of a
7341:
7226:
7070:
7017:
3951:
325:
244:
219:
9877:
9496:
8999:
7122:
4008:
1047:
983:(PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are
701:
9811:
9410:
8963:, is a function that is a solution of a homogeneous linear differential equation with polynomial coefficients.
8260:
In the general case there is no closed-form solution for the homogeneous equation, and one has to use either a
6864:
6386:
3680:
972:
636:
189:
9567:
6160:
The general form of a linear ordinary differential equation of order 1, after dividing out the coefficient of
4813:
There are several methods for solving such an equation. The best method depends on the nature of the function
1938:
1365:
variables. The basic differential operators include the derivative of order 0, which is the identity mapping.
1354:{\displaystyle {\frac {\partial ^{i_{1}+\cdots +i_{n}}}{\partial x_{1}^{i_{1}}\cdots \partial x_{n}^{i_{n}}}}}
361:
9582:
9425:
9415:
6340:
980:
376:
204:
194:
9173:, and try to modify the left side so it becomes a derivative. Specifically, we seek an "integrating factor"
7356:
9405:
9003:
8890:
8141:
7335:
4822:
651:
502:
405:
292:
9670:
9540:
9501:
7421:
3885:
3827:
3640:
3463:
337:
252:
3620:{\displaystyle \left({\frac {d}{dx}}-\alpha \right)\left(x^{k}e^{\alpha x}\right)=kx^{k-1}e^{\alpha x},}
2198:
are arbitrary numbers. Typically, the hypotheses of Carathéodory's theorem are satisfied in an interval
9122:
8702:
8687:
542:
410:
9506:
9262:
Benoit, A., Chyzak, F., Darrasse, A., Gerhold, S., Mezzarobba, M., & Salvy, B. (2010, September).
8400:
3772:
3717:
9791:
9710:
9091:
8724:
8675:
7513:
6435:
For the general non-homogeneous equation, it is useful to multiply both sides of the equation by the
4582:
513:
491:
9846:
9816:
656:
9705:
8694:
8667:
996:
507:
415:
9715:
9685:
9675:
9572:
9127:
9040:
Usefulness of the concept of holonomic functions results of
Zeilberger's theorem, which follows.
9015:
8427:
6334:
2985:
2019:
1181:
1059:
947:
587:
577:
569:
525:
366:
7590:, and this is a different theory. Therefore, the systems that are considered here have the form
5148:
as constants, they can be considered as unknown functions that have to be determined for making
9268:. In International Congress on Mathematical Software (pp. 35-41). Springer, Berlin, Heidelberg.
8713:
6436:
5538:
4888:
3000:
1124:
1004:
400:
27:
Differential equations that are linear with respect to the unknown function and its derivatives
7471:
3007:
of solutions of the differential equation (that is, the kernel of the differential operator).
1007:
allows deciding whether there are solutions in terms of integrals, and computing them if any.
9831:
9826:
9720:
9644:
9623:
9491:
9486:
9481:
9476:
9400:
9379:
9156:
9060:
8712:
Nevertheless, the case of order two with rational coefficients has been completely solved by
7930:
3455:. Thus, applying the differential operator of the equation is equivalent with applying first
1167:
1105:, which does not depend on the unknown function and its derivatives, is sometimes called the
737:
646:
631:
520:
463:
445:
284:
40:
4563:, respectively. This results in a linear system of two linear equations in the two unknowns
9725:
9649:
9618:
8979:
8390:{\displaystyle \mathbf {y} (x)=U(x)\mathbf {y_{0}} +U(x)\int U^{-1}(x)\mathbf {b} (x)\,dx,}
7899:
3687:
2981:
2414:
1770:{\displaystyle L=a_{0}(x)+a_{1}(x){\frac {d}{dx}}+\cdots +a_{n}(x){\frac {d^{n}}{dx^{n}}},}
1609:
1075:
1031:
532:
468:
441:
8086:
8069:
be the homogeneous equation associated to the above matrix equation. Its solutions form a
8:
9735:
9628:
9522:
9095:
9087:
9079:
9048:
3363:, more linearly independent solutions are needed for having a basis. These have the form
1510:{\displaystyle a_{0}(x)+a_{1}(x){\frac {d}{dx}}+\cdots +a_{n}(x){\frac {d^{n}}{dx^{n}}},}
1087:
1071:
564:
549:
450:
314:
153:
120:
111:
3359:, the preceding provides a complete basis of the solutions vector space. In the case of
9856:
9695:
9690:
9613:
9445:
8971:
8956:
8950:
8698:
8182:
5152:
a solution of the non-homogeneous equation. For this purpose, one adds the constraints
4881:
4873:
4400:
3711:
1632:
1381:
1192:
1110:
1015:
984:
559:
554:
437:
9318:
9300:
9282:
9264:
9117:
9112:
9083:
8995:
4287:
1185:
1140:
741:
616:
395:
130:
8666:
A linear ordinary equation of order one with variable coefficients may be solved by
1638:
The language of operators allows a compact writing for differentiable equations: if
671:
9841:
9821:
9022:
9007:
8265:
8261:
2458:, and this allows solving homogeneous linear differential equations rather easily.
2409:
The study of these differential equations with constant coefficients dates back to
1027:
681:
666:
2574:
be a homogeneous linear differential equation with constant coefficients (that is
9801:
9730:
9011:
8991:
4765:{\displaystyle y^{(n)}(x)+a_{1}y^{(n-1)}(x)+\cdots +a_{n-1}y'(x)+a_{n}y(x)=f(x),}
3710:
is also a root, of the same multiplicity. Thus a real basis is obtained by using
1627:(depending on the nature of the functions that are considered). They form also a
1605:
1055:
621:
537:
64:
9806:
9337:
8679:
676:
9796:
9603:
9030:
8878:{\displaystyle x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\cdots +a_{0}y(x)=0,}
8683:
8671:
8134:
6376:
6091:
6087:
3948:
A homogeneous linear differential equation of the second order may be written
2973:
2410:
1624:
1113:), even when this term is a non-constant function. If the constant term is the
1067:
1051:
1023:
1000:
641:
626:
432:
420:
139:
4078:. In all three cases, the general solution depends on two arbitrary constants
3690:. Such a basis may be obtained from the preceding basis by remarking that, if
9871:
9052:
8078:
3505:
3360:
1928:{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}=b(x)}
1569:
1114:
6783:(changing of antiderivative amounts to change the constant of integration).
896:{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)}
9851:
9786:
9700:
9188:
such that multiplying by it makes the left side equal to the derivative of
9056:
8070:
6611:
5534:
3004:
2023:
1616:
1155:
1078:, and numerical evaluation to any precision, with a certified error bound.
661:
611:
497:
125:
8674:. This is not the case for order at least two. This is the main result of
4821:
is a linear combination of exponential and sinusoidal functions, then the
9608:
9348:
8115:
4064:, there are three cases for the solutions, depending on the discriminant
4061:
3428:. For proving that these functions are solutions, one may remark that if
3356:
1628:
1620:
729:
69:
9253:. Journal of computational and applied mathematics. 32.3 (1990): 321-368
9059:
is holonomic, then the series defines a holonomic function (even if the
4392:{\displaystyle c_{1}e^{(\alpha +\beta i)x}+c_{2}e^{(\alpha -\beta i)x},}
9836:
9075:
9026:
8967:
1246:
1019:
1011:
686:
4592:
9598:
9034:
8975:
7416:
appear in an equation, one may replace them by new unknown functions
6086:, and their derivatives). This system can be solved by any method of
1035:
427:
148:
91:
81:
32:
Partial differential equation § Linear equations of second order
8693:
The impossibility of solving by quadrature can be compared with the
4494:{\displaystyle e^{\alpha x}(c_{1}\cos(\beta x)+c_{2}\sin(\beta x)).}
9638:
9071:
4880:
satisfies a homogeneous linear differential equation, typically, a
1063:
745:
6155:
6036:{\displaystyle f=u'_{1}y_{1}^{(n-1)}+\cdots +u'_{n}y_{n}^{(n-1)}.}
5910:
and its derivatives by these expressions, and using the fact that
2266:
9342:
9276:
9074:
operations can be done automatically on these functions, such as
5641:{\displaystyle y^{(i)}=u_{1}y_{1}^{(i)}+\cdots +u_{n}y_{n}^{(i)}}
5010:{\displaystyle y^{(n)}+a_{1}y^{(n-1)}+\cdots +a_{n-1}y'+a_{n}y=0}
2866:
of the differential equation, which is the left-hand side of the
102:
97:
86:
1010:
The solutions of homogeneous linear differential equations with
8987:
7014:
Dividing the original equation by one of these solutions gives
4868:
a constant (which need not be the same in each term), then the
1143:
appear as coefficients in the associated homogeneous equation.
1043:
9345:. Automatic and interactive study of many holonomic functions.
9098:
at infinity and near singularities, proof of identities, etc.
8670:, which means that the solutions may be expressed in terms of
5928:
are solutions of the original homogeneous equation, one gets
4589:
for the solution of the DEQ and its derivative are specified.
2567:{\displaystyle a_{0}y+a_{1}y'+a_{2}y''+\cdots +a_{n}y^{(n)}=0}
2381:{\displaystyle a_{0}y+a_{1}y'+a_{2}y''+\cdots +a_{n}y^{(n)}=0}
999:, which means that the solutions may be expressed in terms of
6601:{\displaystyle -fe^{-F}={\tfrac {d}{dx}}\left(e^{-F}\right),}
4894:
The general solution of the associated homogeneous equation
2963:{\displaystyle a_{0}+a_{1}t+a_{2}t^{2}+\cdots +a_{n}t^{n}=0.}
2173:{\displaystyle S_{0}(x)+c_{1}S_{1}(x)+\cdots +c_{n}S_{n}(x),}
9250:
A holonomic systems approach to special functions identities
7966:. In matrix notation, this system may be written (omitting "
6687:{\displaystyle {\frac {d}{dx}}\left(ye^{-F}\right)=ge^{-F}.}
6383:. Thus, the general solution of the homogeneous equation is
4107:, the characteristic polynomial has two distinct real roots
2026:
of the solutions of the (homogeneous) differential equation
8983:
3432:
is a root of the characteristic polynomial of multiplicity
3414:
is a root of the characteristic polynomial of multiplicity
1039:
8661:
8275:, the general solution of the non-homogeneous equation is
7329:
4541:, one equates the values of the above general solution at
2859:{\displaystyle a_{0}+a_{1}t+a_{2}t^{2}+\cdots +a_{n}t^{n}}
2036:
In the case of an ordinary differential operator of order
1544:
are differentiable functions, and the nonnegative integer
1018:. This class of functions is stable under sums, products,
8489:
the solution that satisfies these initial conditions is
8016:{\displaystyle \mathbf {y} '=A\mathbf {y} +\mathbf {b} .}
4581:. Solving this system gives the solution for a so-called
3355:
In the case where the characteristic polynomial has only
2595:
Searching solutions of this equation that have the form
2044:
implies that, under very mild conditions, the kernel of
1090:
that appears in a (linear) differential equation is the
744:
in the unknown function and its derivatives, that is an
9265:
The dynamic dictionary of mathematical functions (DDMF)
6444:
of a solution of the homogeneous equation. This gives
9051:
with polynomial coefficients. The coefficients of the
8145:
6558:
6350:
3643:
3466:
1777:
is a linear differential operator, then the equation
1579:
be a linear differential operator. The application of
1191:, or, in the case of several variables, to one of its
960:
are the successive derivatives of an unknown function
8893:
8733:
8495:
8482:{\displaystyle \mathbf {y} (x_{0})=\mathbf {y} _{0},}
8439:
8403:
8281:
8195:
8144:
8089:
8034:
7980:
7933:
7902:
7596:
7516:
7474:
7424:
7359:
7267:
7229:
7170:
7125:
7073:
7020:
6982:
6931:
6867:
6797:
6700:
6620:
6534:
6450:
6389:
6343:
6269:
6177:
5934:
5665:
5547:
5158:
5023:
4900:
4607:
4409:
4306:
4207:
4121:
4011:
3954:
3888:
3830:
3775:
3720:
3514:
3369:
3282:
3198:
3103:
3024:
2875:
2777:
2611:
2467:
2281:
2077:
1941:
1783:
1644:
1390:
1255:
1205:
754:
9047:
is a sequence of numbers that may be generated by a
6322:{\displaystyle {\frac {y'}{y}}=f,\qquad \log y=k+F,}
5112:
is a basis of the vector space of the solutions and
4593:
Non-homogeneous equation with constant coefficients
4177:{\displaystyle c_{1}e^{\alpha x}+c_{2}e^{\beta x}.}
3436:, the characteristic polynomial may be factored as
9315:An Introduction to Ordinary Differential Equations
9312:
8931:
8877:
8651:
8481:
8418:
8389:
8252:
8167:
8104:
8061:
8015:
7952:
7915:
7886:
7554:
7498:
7456:
7406:
7320:{\displaystyle y(x)=x^{2}+{\frac {\alpha -1}{x}}.}
7319:
7253:
7215:
7156:
7112:
7059:
7004:
6968:
6917:
6853:
6763:
6686:
6600:
6518:
6414:
6367:
6321:
6242:
6035:
5896:
5640:
5525:
5084:
5009:
4802:is the unknown function (for sake of simplicity, "
4764:
4493:
4391:
4269:
4193:, the characteristic polynomial has a double root
4176:
4042:
3997:
3932:
3874:
3816:
3761:
3667:
3619:
3490:
3398:
3341:
3269:{\displaystyle e^{ix},\;e^{-ix},\;e^{x},\;xe^{x}.}
3268:
3169:
3089:
2962:
2858:
2750:
2566:
2380:
2172:
1968:
1927:
1769:
1608:, since it maps sums to sums and the product by a
1509:
1353:
1237:
895:
9766:List of nonlinear ordinary differential equations
9294:
9277:Birkhoff, Garrett & Rota, Gian-Carlo (1978),
8129:is a matrix of constants, or, more generally, if
4829:is a linear combination of functions of the form
3700:is a root of the characteristic polynomial, then
3342:{\displaystyle \cos x,\;\sin x,\;e^{x},\;xe^{x}.}
9869:
9771:List of nonlinear partial differential equations
8253:{\displaystyle {\frac {d}{dx}}\exp(B)=A\exp(B).}
6969:{\displaystyle {\frac {y'}{y}}=-{\frac {1}{x}},}
5085:{\displaystyle y=u_{1}y_{1}+\cdots +u_{n}y_{n},}
1982:and the right-hand and of the equation, such as
6156:First-order equation with variable coefficients
2422:, which is the unique solution of the equation
2271:A homogeneous linear differential equation has
2267:Homogeneous equation with constant coefficients
1161:
1117:, then the differential equation is said to be
9761:List of linear ordinary differential equations
9317:, Cambridge, UK.: Cambridge University Press,
9299:, Cambridge, UK.: Cambridge University Press,
7578:A linear system of the first order, which has
9364:
9338:http://eqworld.ipmnet.ru/en/solutions/ode.htm
6764:{\displaystyle y=ce^{F}+e^{F}\int ge^{-F}dx,}
3192:(multiplicity 2). The solution basis is thus
709:
9243:
9241:
9239:
8062:{\displaystyle \mathbf {u} '=A\mathbf {u} .}
4817:that makes the equation non-homogeneous. If
4399:which may be rewritten in real terms, using
9343:Dynamic Dictionary of Mathematical Function
8686:, and whose recent developments are called
6854:{\displaystyle y'(x)+{\frac {y(x)}{x}}=3x.}
3170:{\displaystyle z^{4}-2z^{3}+2z^{2}-2z+1=0.}
9378:
9371:
9357:
7739:
7738:
6519:{\displaystyle y'e^{-F}-yfe^{-F}=ge^{-F}.}
5366:
5365:
4601:with constant coefficients may be written
3322:
3308:
3295:
3249:
3235:
3215:
716:
702:
9236:
8639:
8377:
6918:{\displaystyle y'(x)+{\frac {y(x)}{x}}=0}
6357:
2601:is equivalent to searching the constants
2052:, and that the solutions of the equation
2018:of a linear differential operator is its
8719:
6073:whose coefficients are known functions (
4286:, the characteristic polynomial has two
4270:{\displaystyle (c_{1}+c_{2}x)e^{-ax/2}.}
4115:. In this case, the general solution is
3090:{\displaystyle y''''-2y'''+2y''-2y'+y=0}
1026:, and contains many usual functions and
18:First-order linear differential equation
9281:, New York: John Wiley and Sons, Inc.,
8662:Higher order with variable coefficients
7330:System of linear differential equations
6368:{\displaystyle F=\textstyle \int f\,dx}
1238:{\displaystyle {\frac {d^{i}}{dx^{i}}}}
14:
9870:
8944:
7407:{\displaystyle y',y'',\ldots ,y^{(k)}}
6045:This equation and the above ones with
2228:, and there is a positive real number
9352:
8932:{\displaystyle a_{0},\ldots ,a_{n-1}}
8264:, or an approximation method such as
8168:{\displaystyle \textstyle B=\int Adx}
8077:, and are therefore the columns of a
6252:If the equation is homogeneous, i.e.
4872:may be used. Still more general, the
4810:" will be omitted in the following).
4005:and its characteristic polynomial is
3504:as characteristic polynomial. By the
9756:List of named differential equations
8189:. In fact, in these cases, one has
6861:The associated homogeneous equation
6243:{\displaystyle y'(x)=f(x)y(x)+g(x).}
4597:A non-homogeneous equation of order
3943:
3668:{\textstyle {\frac {d}{dx}}-\alpha }
3491:{\textstyle {\frac {d}{dx}}-\alpha }
2988:of the values of these solutions at
1081:
990:
164:List of named differential equations
9681:Method of undetermined coefficients
9462:Dependent and independent variables
9297:The Nature of Mathematical Modeling
9159:technique we write the equation as
8433:If initial conditions are given as
7457:{\displaystyle y_{1},\ldots ,y_{k}}
6049:as left-hand side form a system of
5906:Replacing in the original equation
4870:method of undetermined coefficients
3933:{\displaystyle x^{k}e^{ax}\sin(bx)}
3875:{\displaystyle x^{k}e^{ax}\cos(bx)}
1612:to the product by the same scalar.
950:that do not need to be linear, and
237:Dependent and independent variables
24:
6775:is a constant of integration, and
3399:{\displaystyle x^{k}e^{\alpha x},}
2764:(which is never zero), shows that
1372:(abbreviated, in this article, as
1323:
1295:
1259:
25:
9889:
9331:
9025:; in particular, sums, products,
9021:Holonomic functions have several
7261:one gets the particular solution
7005:{\displaystyle y={\frac {c}{x}}.}
6614:allows rewriting the equation as
6263:, one may rewrite and integrate:
4825:may be used. If, more generally,
3276:A real basis of solution is thus
2022:as a linear mapping, that is the
1109:of the equation (by analogy with
9578:Carathéodory's existence theorem
8626:
8560:
8556:
8497:
8466:
8441:
8419:{\displaystyle \mathbf {y_{0}} }
8410:
8406:
8364:
8317:
8313:
8283:
8052:
8037:
8006:
7998:
7983:
7466:that must satisfy the equations
7342:system of differential equations
3817:{\displaystyle x^{k}e^{(a-ib)x}}
3762:{\displaystyle x^{k}e^{(a+ib)x}}
3097:has the characteristic equation
2042:Carathéodory's existence theorem
372:Carathéodory's existence theorem
9279:Ordinary Differential Equations
7555:{\displaystyle y_{i}'=y_{i+1},}
7216:{\displaystyle y(x)=x^{2}+c/x.}
6294:
4887:The most general method is the
3500:and then the operator that has
2048:is a vector space of dimension
1131:associated homogeneous equation
1048:inverse trigonometric functions
9256:
9149:
9140:
8863:
8857:
8832:
8826:
8821:
8809:
8766:
8760:
8755:
8749:
8636:
8630:
8622:
8616:
8578:
8572:
8551:
8538:
8522:
8516:
8507:
8501:
8458:
8445:
8374:
8368:
8360:
8354:
8335:
8329:
8308:
8302:
8293:
8287:
8244:
8238:
8223:
8217:
8099:
8093:
7864:
7858:
7820:
7814:
7792:
7786:
7766:
7760:
7718:
7712:
7674:
7668:
7646:
7640:
7620:
7614:
7399:
7393:
7277:
7271:
7239:
7233:
7180:
7174:
7084:
7074:
6900:
6894:
6882:
6876:
6830:
6824:
6812:
6806:
6694:Thus, the general solution is
6234:
6228:
6219:
6213:
6207:
6201:
6192:
6186:
6025:
6013:
5976:
5964:
5886:
5874:
5837:
5825:
5794:
5782:
5751:
5745:
5711:
5705:
5677:
5671:
5633:
5627:
5593:
5587:
5559:
5553:
5511:
5499:
5462:
5450:
5419:
5407:
4947:
4935:
4912:
4906:
4864:is a nonnegative integer, and
4756:
4750:
4741:
4735:
4716:
4710:
4674:
4668:
4663:
4651:
4630:
4624:
4619:
4613:
4485:
4482:
4473:
4451:
4442:
4423:
4378:
4363:
4337:
4322:
4300:, and the general solution is
4237:
4208:
4201:, and the general solution is
3927:
3918:
3869:
3860:
3806:
3791:
3751:
3736:
3681:fundamental theorem of algebra
2592:are real or complex numbers).
2553:
2547:
2367:
2361:
2164:
2158:
2126:
2120:
2094:
2088:
1960:
1954:
1922:
1916:
1905:
1899:
1891:
1885:
1855:
1849:
1825:
1819:
1800:
1794:
1732:
1726:
1689:
1683:
1667:
1661:
1472:
1466:
1429:
1423:
1407:
1401:
1135:. A differential equation has
973:ordinary differential equation
890:
884:
873:
867:
859:
853:
826:
820:
796:
790:
771:
765:
459: / Integral solutions
13:
1:
9133:
9108:Continuous-repayment mortgage
8118:is not the zero function. If
7588:differential-algebraic system
7254:{\displaystyle y(1)=\alpha ,}
7113:{\displaystyle (xy)'=3x^{2},}
7060:{\displaystyle xy'+y=3x^{2}.}
4790:are real or complex numbers,
3998:{\displaystyle y''+ay'+by=0,}
3629:and thus one gets zero after
1635:of differentiable functions.
981:partial differential equation
9406:Notation for differentiation
9004:inverse hyperbolic functions
7336:Matrix differential equation
4823:exponential response formula
4545:and its derivative there to
1370:linear differential operator
1361:in the case of functions of
1162:Linear differential operator
977:linear differential equation
734:linear differential equation
503:Exponential response formula
249:Coupled / Decoupled
7:
9502:Exact differential equation
9313:Robinson, James C. (2004),
9101:
8941:are constant coefficients.
7157:{\displaystyle xy=x^{3}+c,}
4891:, which is presented here.
4043:{\displaystyle r^{2}+ar+b.}
1180:is a mapping that maps any
1174:basic differential operator
10:
9894:
9295:Gershenfeld, Neil (1999),
9155:Motivation: In analogy to
9123:Linear difference equation
8948:
8703:differential Galois theory
8688:differential Galois theory
7339:
7333:
7223:For the initial condition
6786:
6432:is an arbitrary constant.
3410:is a nonnegative integer,
1199:. It is commonly denoted
1165:
1094:of the equation. The term
29:
9812:Józef Maria Hoene-Wroński
9792:Gottfried Wilhelm Leibniz
9779:
9748:
9658:
9591:
9583:Cauchy–Kowalevski theorem
9560:
9553:
9515:
9454:
9393:
9386:
6779:is any antiderivative of
6415:{\displaystyle y=ce^{F},}
4585:, in which the values at
3506:exponential shift theorem
2972:When these roots are all
2770:characteristic polynomial
1066:, such as computation of
637:Józef Maria Hoene-Wroński
583:Undetermined coefficients
492:Method of characteristics
377:Cauchy–Kowalevski theorem
9706:Finite difference method
9016:hypergeometric functions
8397:where the column matrix
7499:{\displaystyle y'=y_{1}}
1969:{\displaystyle Ly=b(x).}
1060:hypergeometric functions
1014:coefficients are called
948:differentiable functions
362:Picard–Lindelöf theorem
356:Existence and uniqueness
9686:Variation of parameters
9676:Separation of variables
9573:Peano existence theorem
9568:Picard–Lindelöf theorem
9455:Attributes of variables
9128:Variation of parameters
8697:, which states that an
8678:which was initiated by
8428:constant of integration
7953:{\displaystyle a_{i,j}}
6335:constant of integration
4794:is a given function of
2999:. Together they form a
2986:Vandermonde determinant
2868:characteristic equation
1182:differentiable function
971:Such an equation is an
588:Variation of parameters
578:Separation of variables
367:Peano existence theorem
9878:Differential equations
9847:Carl David Tolmé Runge
9421:Differential-algebraic
9380:Differential equations
8933:
8879:
8725:Cauchy–Euler equations
8653:
8483:
8420:
8391:
8254:
8177:, then one may choose
8169:
8106:
8063:
8017:
7954:
7917:
7888:
7582:unknown functions and
7556:
7500:
7458:
7408:
7321:
7255:
7217:
7158:
7114:
7061:
7006:
6970:
6919:
6855:
6765:
6688:
6602:
6520:
6416:
6369:
6323:
6244:
6037:
5898:
5642:
5527:
5086:
5011:
4889:variation of constants
4766:
4495:
4393:
4271:
4178:
4044:
3999:
3934:
3876:
3818:
3763:
3669:
3621:
3492:
3400:
3343:
3270:
3171:
3091:
2964:
2860:
2768:must be a root of the
2752:
2568:
2439:. It follows that the
2382:
2174:
1970:
1929:
1771:
1511:
1355:
1239:
1125:homogeneous polynomial
897:
657:Carl David Tolmé Runge
200:Differential-algebraic
41:Differential equations
9832:Augustin-Louis Cauchy
9827:Joseph-Louis Lagrange
9721:Finite element method
9711:Crank–Nicolson method
9645:Numerical integration
9624:Exponential stability
9516:Relation to processes
9401:Differential operator
9157:completing the square
9146:Gershenfeld 1999, p.9
9061:radius of convergence
9000:inverse trigonometric
8934:
8880:
8720:Cauchy–Euler equation
8676:Picard–Vessiot theory
8654:
8484:
8421:
8392:
8255:
8170:
8107:
8064:
8018:
7955:
7918:
7916:{\displaystyle b_{n}}
7889:
7557:
7501:
7459:
7409:
7322:
7256:
7218:
7159:
7115:
7062:
7007:
6971:
6920:
6856:
6791:Solving the equation
6766:
6689:
6603:
6521:
6417:
6370:
6324:
6245:
6090:. The computation of
6038:
5899:
5643:
5528:
5087:
5012:
4767:
4504:Finding the solution
4496:
4394:
4272:
4179:
4045:
4000:
3935:
3877:
3819:
3764:
3688:real-valued functions
3670:
3622:
3493:
3401:
3344:
3271:
3172:
3092:
2984:, by considering the
2965:
2861:
2753:
2569:
2413:, who introduced the
2383:
2273:constant coefficients
2175:
1971:
1930:
1772:
1512:
1356:
1240:
1168:Differential operator
1137:constant coefficients
979:may also be a linear
898:
740:that is defined by a
738:differential equation
647:Augustin-Louis Cauchy
632:Joseph-Louis Lagrange
464:Numerical integration
446:Exponential stability
309:Relation to processes
9726:Finite volume method
9650:Dirac delta function
9619:Asymptotic stability
9561:Existence/uniqueness
9426:Integro-differential
8980:exponential function
8891:
8731:
8695:Abel–Ruffini theorem
8493:
8437:
8401:
8279:
8193:
8142:
8105:{\displaystyle U(x)}
8087:
8032:
7978:
7931:
7900:
7594:
7514:
7472:
7422:
7357:
7265:
7227:
7168:
7123:
7071:
7018:
6980:
6929:
6865:
6795:
6698:
6618:
6532:
6448:
6387:
6341:
6267:
6175:
6053:linear equations in
5932:
5663:
5545:
5156:
5021:
4898:
4605:
4407:
4304:
4205:
4119:
4009:
3952:
3886:
3828:
3773:
3718:
3641:
3512:
3464:
3367:
3280:
3196:
3101:
3022:
2982:linearly independent
2873:
2775:
2609:
2465:
2415:exponential function
2279:
2075:
1939:
1781:
1642:
1552:of the operator (if
1388:
1253:
1203:
1076:asymptotic expansion
1032:exponential function
752:
469:Dirac delta function
205:Integro-differential
9736:Perturbation theory
9716:Runge–Kutta methods
9696:Integral transforms
9629:Rate of convergence
9525:(discrete analogue)
9247:Zeilberger, Doron.
9096:asymptotic behavior
9049:recurrence relation
8972:algebraic functions
8945:Holonomic functions
8714:Kovacic's algorithm
8602:
8271:Knowing the matrix
7759:
7613:
7529:
6029:
6002:
5980:
5953:
5890:
5863:
5841:
5814:
5798:
5771:
5755:
5715:
5637:
5597:
5515:
5488:
5466:
5439:
5423:
5396:
5357:
5344:
5322:
5309:
5293:
5280:
5243:
5211:
5185:
3459:times the operator
2275:if it has the form
2202:, if the functions
1587:is usually denoted
1347:
1319:
1193:partial derivatives
1111:algebraic equations
1088:order of derivation
1016:holonomic functions
1005:Kovacic's algorithm
985:partial derivatives
565:Perturbation theory
560:Integral transforms
451:Rate of convergence
317:(discrete analogue)
154:Population dynamics
121:Continuum mechanics
112:Applied mathematics
9857:Sofya Kovalevskaya
9691:Integrating factor
9614:Lyapunov stability
9534:Stochastic partial
9090:, localization of
9045:holonomic sequence
9023:closure properties
8957:holonomic function
8951:holonomic function
8929:
8875:
8699:algebraic equation
8649:
8581:
8479:
8416:
8387:
8250:
8165:
8164:
8133:commutes with its
8102:
8059:
8013:
7950:
7913:
7884:
7882:
7747:
7601:
7552:
7517:
7496:
7454:
7404:
7317:
7251:
7213:
7154:
7110:
7057:
7002:
6966:
6915:
6851:
6761:
6684:
6598:
6572:
6516:
6412:
6365:
6364:
6319:
6240:
6033:
6003:
5990:
5954:
5941:
5894:
5864:
5851:
5815:
5802:
5772:
5759:
5735:
5695:
5638:
5617:
5577:
5523:
5521:
5489:
5476:
5440:
5427:
5397:
5384:
5345:
5332:
5310:
5297:
5281:
5268:
5231:
5199:
5173:
5082:
5007:
4882:holonomic function
4874:annihilator method
4762:
4491:
4389:
4267:
4174:
4040:
3995:
3930:
3872:
3814:
3759:
3665:
3617:
3488:
3396:
3339:
3266:
3167:
3087:
2960:
2856:
2748:
2564:
2378:
2224:are continuous in
2170:
1966:
1925:
1767:
1507:
1382:linear combination
1351:
1326:
1298:
1235:
1141:constant functions
893:
555:Integrating factor
396:Initial conditions
331:Stochastic partial
9865:
9864:
9744:
9743:
9549:
9548:
9306:978-0-521-57095-4
9233:, as in the text.
9118:Laplace transform
9113:Fourier transform
9084:definite integral
9008:special functions
8996:hyperbolic cosine
8961:D-finite function
8959:, also called a
8209:
7962:are functions of
7312:
6997:
6961:
6945:
6907:
6837:
6634:
6571:
6283:
4288:complex conjugate
3944:Second-order case
3657:
3533:
3480:
3353:
3352:
2443:th derivative of
1935:may be rewritten
1762:
1705:
1502:
1445:
1349:
1233:
1082:Basic terminology
1028:special functions
991:Types of solution
742:linear polynomial
726:
725:
617:Gottfried Leibniz
508:Finite difference
300:
299:
161:
160:
131:Dynamical systems
16:(Redirected from
9885:
9842:Phyllis Nicolson
9822:Rudolf Lipschitz
9659:Solution methods
9634:Series solutions
9558:
9557:
9391:
9390:
9373:
9366:
9359:
9350:
9349:
9327:
9309:
9291:
9269:
9260:
9254:
9245:
9234:
9232:
9218:
9208:
9193:
9187:
9172:
9153:
9147:
9144:
9012:Bessel functions
8940:
8938:
8936:
8935:
8930:
8928:
8927:
8903:
8902:
8884:
8882:
8881:
8876:
8853:
8852:
8825:
8824:
8803:
8802:
8787:
8786:
8759:
8758:
8743:
8742:
8658:
8656:
8655:
8650:
8629:
8615:
8614:
8601:
8596:
8595:
8594:
8565:
8564:
8563:
8550:
8549:
8537:
8536:
8500:
8488:
8486:
8485:
8480:
8475:
8474:
8469:
8457:
8456:
8444:
8426:is an arbitrary
8425:
8423:
8422:
8417:
8415:
8414:
8413:
8396:
8394:
8393:
8388:
8367:
8353:
8352:
8322:
8321:
8320:
8286:
8274:
8266:Magnus expansion
8262:numerical method
8259:
8257:
8256:
8251:
8210:
8208:
8197:
8188:
8180:
8176:
8174:
8172:
8171:
8166:
8132:
8128:
8124:
8113:
8111:
8109:
8108:
8103:
8076:
8068:
8066:
8065:
8060:
8055:
8044:
8040:
8022:
8020:
8019:
8014:
8009:
8001:
7990:
7986:
7973:
7965:
7961:
7959:
7957:
7956:
7951:
7949:
7948:
7924:
7922:
7920:
7919:
7914:
7912:
7911:
7893:
7891:
7890:
7885:
7883:
7876:
7875:
7857:
7856:
7832:
7831:
7813:
7812:
7785:
7784:
7755:
7734:
7730:
7729:
7711:
7710:
7686:
7685:
7667:
7666:
7639:
7638:
7609:
7585:
7581:
7574:
7563:
7561:
7559:
7558:
7553:
7548:
7547:
7525:
7507:
7505:
7503:
7502:
7497:
7495:
7494:
7482:
7465:
7463:
7461:
7460:
7455:
7453:
7452:
7434:
7433:
7415:
7413:
7411:
7410:
7405:
7403:
7402:
7378:
7367:
7326:
7324:
7323:
7318:
7313:
7308:
7297:
7292:
7291:
7260:
7258:
7257:
7252:
7222:
7220:
7219:
7214:
7206:
7195:
7194:
7163:
7161:
7160:
7155:
7144:
7143:
7119:
7117:
7116:
7111:
7106:
7105:
7090:
7066:
7064:
7063:
7058:
7053:
7052:
7031:
7011:
7009:
7008:
7003:
6998:
6990:
6975:
6973:
6972:
6967:
6962:
6954:
6946:
6941:
6933:
6924:
6922:
6921:
6916:
6908:
6903:
6889:
6875:
6860:
6858:
6857:
6852:
6838:
6833:
6819:
6805:
6782:
6778:
6774:
6770:
6768:
6767:
6762:
6751:
6750:
6732:
6731:
6719:
6718:
6693:
6691:
6690:
6685:
6680:
6679:
6661:
6657:
6656:
6655:
6635:
6633:
6622:
6609:
6607:
6605:
6604:
6599:
6594:
6590:
6589:
6573:
6570:
6559:
6553:
6552:
6525:
6523:
6522:
6517:
6512:
6511:
6493:
6492:
6471:
6470:
6458:
6443:
6431:
6421:
6419:
6418:
6413:
6408:
6407:
6382:
6374:
6372:
6371:
6366:
6333:is an arbitrary
6332:
6328:
6326:
6325:
6320:
6284:
6279:
6271:
6262:
6249:
6247:
6246:
6241:
6185:
6170:
6147:
6111:
6085:
6076:
6072:
6052:
6048:
6042:
6040:
6039:
6034:
6028:
6011:
5998:
5979:
5962:
5949:
5927:
5909:
5903:
5901:
5900:
5895:
5889:
5872:
5859:
5840:
5823:
5810:
5797:
5780:
5767:
5754:
5743:
5734:
5733:
5714:
5703:
5694:
5693:
5681:
5680:
5658:
5647:
5645:
5644:
5639:
5636:
5625:
5616:
5615:
5596:
5585:
5576:
5575:
5563:
5562:
5533:which imply (by
5532:
5530:
5529:
5524:
5522:
5514:
5497:
5484:
5465:
5448:
5435:
5422:
5405:
5392:
5361:
5353:
5340:
5318:
5305:
5289:
5276:
5253:
5252:
5239:
5221:
5220:
5207:
5195:
5194:
5181:
5151:
5147:
5129:
5111:
5091:
5089:
5088:
5083:
5078:
5077:
5068:
5067:
5049:
5048:
5039:
5038:
5016:
5014:
5013:
5008:
4997:
4996:
4984:
4976:
4975:
4951:
4950:
4929:
4928:
4916:
4915:
4879:
4867:
4863:
4859:
4848:
4837:
4828:
4820:
4816:
4809:
4801:
4797:
4793:
4789:
4771:
4769:
4768:
4763:
4731:
4730:
4709:
4701:
4700:
4667:
4666:
4645:
4644:
4623:
4622:
4600:
4588:
4580:
4571:
4562:
4553:
4544:
4540:
4527:
4514:
4500:
4498:
4497:
4492:
4466:
4465:
4435:
4434:
4422:
4421:
4398:
4396:
4395:
4390:
4385:
4384:
4357:
4356:
4344:
4343:
4316:
4315:
4299:
4285:
4276:
4274:
4273:
4268:
4263:
4262:
4258:
4233:
4232:
4220:
4219:
4200:
4192:
4183:
4181:
4180:
4175:
4170:
4169:
4157:
4156:
4144:
4143:
4131:
4130:
4114:
4110:
4106:
4095:
4086:
4077:
4059:
4055:
4049:
4047:
4046:
4041:
4021:
4020:
4004:
4002:
4001:
3996:
3976:
3962:
3939:
3937:
3936:
3931:
3911:
3910:
3898:
3897:
3881:
3879:
3878:
3873:
3853:
3852:
3840:
3839:
3823:
3821:
3820:
3815:
3813:
3812:
3785:
3784:
3768:
3766:
3765:
3760:
3758:
3757:
3730:
3729:
3714:, and replacing
3709:
3699:
3676:
3674:
3672:
3671:
3666:
3658:
3656:
3645:
3635:
3626:
3624:
3623:
3618:
3613:
3612:
3600:
3599:
3578:
3574:
3573:
3572:
3560:
3559:
3545:
3541:
3534:
3532:
3521:
3503:
3499:
3497:
3495:
3494:
3489:
3481:
3479:
3468:
3458:
3454:
3435:
3431:
3427:
3417:
3413:
3409:
3405:
3403:
3402:
3397:
3392:
3391:
3379:
3378:
3348:
3346:
3345:
3340:
3335:
3334:
3318:
3317:
3275:
3273:
3272:
3267:
3262:
3261:
3245:
3244:
3231:
3230:
3211:
3210:
3191:
3187:
3180:
3177:This has zeros,
3176:
3174:
3173:
3168:
3145:
3144:
3129:
3128:
3113:
3112:
3096:
3094:
3093:
3088:
3074:
3060:
3046:
3032:
3010:
3009:
2998:
2979:
2969:
2967:
2966:
2961:
2953:
2952:
2943:
2942:
2924:
2923:
2914:
2913:
2898:
2897:
2885:
2884:
2865:
2863:
2862:
2857:
2855:
2854:
2845:
2844:
2826:
2825:
2816:
2815:
2800:
2799:
2787:
2786:
2767:
2763:
2757:
2755:
2754:
2749:
2741:
2740:
2728:
2727:
2718:
2717:
2699:
2698:
2686:
2685:
2676:
2675:
2663:
2662:
2647:
2646:
2634:
2633:
2621:
2620:
2604:
2600:
2591:
2573:
2571:
2570:
2565:
2557:
2556:
2541:
2540:
2522:
2514:
2513:
2501:
2493:
2492:
2477:
2476:
2457:
2448:
2442:
2438:
2431:
2421:
2405:
2387:
2385:
2384:
2379:
2371:
2370:
2355:
2354:
2336:
2328:
2327:
2315:
2307:
2306:
2291:
2290:
2262:
2258:
2254:
2249:
2231:
2227:
2223:
2201:
2197:
2179:
2177:
2176:
2171:
2157:
2156:
2147:
2146:
2119:
2118:
2109:
2108:
2087:
2086:
2070:
2051:
2047:
2039:
2032:
2010:
2000:
1981:
1975:
1973:
1972:
1967:
1934:
1932:
1931:
1926:
1909:
1908:
1884:
1883:
1865:
1848:
1847:
1835:
1818:
1817:
1793:
1792:
1776:
1774:
1773:
1768:
1763:
1761:
1760:
1759:
1746:
1745:
1736:
1725:
1724:
1706:
1704:
1693:
1682:
1681:
1660:
1659:
1603:
1592:
1586:
1582:
1578:
1567:
1547:
1543:
1516:
1514:
1513:
1508:
1503:
1501:
1500:
1499:
1486:
1485:
1476:
1465:
1464:
1446:
1444:
1433:
1422:
1421:
1400:
1399:
1364:
1360:
1358:
1357:
1352:
1350:
1348:
1346:
1345:
1344:
1334:
1318:
1317:
1316:
1306:
1293:
1292:
1291:
1290:
1272:
1271:
1257:
1249:functions, and
1244:
1242:
1241:
1236:
1234:
1232:
1231:
1230:
1217:
1216:
1207:
1198:
1188:
1179:
1152:
1151:
1133:
1132:
1104:
1056:Bessel functions
967:
964:of the variable
963:
959:
945:
934:
933:
917:
902:
900:
899:
894:
877:
876:
852:
851:
836:
819:
818:
806:
789:
788:
764:
763:
718:
711:
704:
682:Phyllis Nicolson
667:Rudolf Lipschitz
550:Green's function
526:Infinite element
517:
482:Solution methods
460:
318:
229:By variable type
183:
182:
65:Natural sciences
58:
57:
37:
36:
21:
9893:
9892:
9888:
9887:
9886:
9884:
9883:
9882:
9868:
9867:
9866:
9861:
9802:Jacob Bernoulli
9775:
9740:
9731:Galerkin method
9654:
9592:Solution topics
9587:
9545:
9511:
9450:
9382:
9377:
9334:
9325:
9307:
9289:
9273:
9272:
9261:
9257:
9246:
9237:
9220:
9210:
9195:
9189:
9174:
9160:
9154:
9150:
9145:
9141:
9136:
9104:
8992:hyperbolic sine
8953:
8947:
8917:
8913:
8898:
8894:
8892:
8889:
8888:
8886:
8848:
8844:
8808:
8804:
8792:
8788:
8776:
8772:
8748:
8744:
8738:
8734:
8732:
8729:
8728:
8722:
8664:
8625:
8607:
8603:
8597:
8590:
8586:
8585:
8559:
8555:
8554:
8545:
8541:
8529:
8525:
8496:
8494:
8491:
8490:
8470:
8465:
8464:
8452:
8448:
8440:
8438:
8435:
8434:
8409:
8405:
8404:
8402:
8399:
8398:
8363:
8345:
8341:
8316:
8312:
8311:
8282:
8280:
8277:
8276:
8272:
8201:
8196:
8194:
8191:
8190:
8186:
8178:
8143:
8140:
8139:
8137:
8130:
8126:
8119:
8088:
8085:
8084:
8082:
8074:
8051:
8036:
8035:
8033:
8030:
8029:
8005:
7997:
7982:
7981:
7979:
7976:
7975:
7967:
7963:
7938:
7934:
7932:
7929:
7928:
7926:
7907:
7903:
7901:
7898:
7897:
7895:
7881:
7880:
7871:
7867:
7846:
7842:
7827:
7823:
7802:
7798:
7780:
7776:
7769:
7751:
7744:
7743:
7732:
7731:
7725:
7721:
7700:
7696:
7681:
7677:
7656:
7652:
7634:
7630:
7623:
7605:
7597:
7595:
7592:
7591:
7583:
7579:
7565:
7537:
7533:
7521:
7515:
7512:
7511:
7509:
7490:
7486:
7475:
7473:
7470:
7469:
7467:
7448:
7444:
7429:
7425:
7423:
7420:
7419:
7417:
7392:
7388:
7371:
7360:
7358:
7355:
7354:
7352:
7344:
7338:
7332:
7298:
7296:
7287:
7283:
7266:
7263:
7262:
7228:
7225:
7224:
7202:
7190:
7186:
7169:
7166:
7165:
7139:
7135:
7124:
7121:
7120:
7101:
7097:
7083:
7072:
7069:
7068:
7048:
7044:
7024:
7019:
7016:
7015:
6989:
6981:
6978:
6977:
6953:
6934:
6932:
6930:
6927:
6926:
6890:
6888:
6868:
6866:
6863:
6862:
6820:
6818:
6798:
6796:
6793:
6792:
6789:
6780:
6776:
6772:
6743:
6739:
6727:
6723:
6714:
6710:
6699:
6696:
6695:
6672:
6668:
6648:
6644:
6640:
6636:
6626:
6621:
6619:
6616:
6615:
6582:
6578:
6574:
6563:
6557:
6545:
6541:
6533:
6530:
6529:
6527:
6504:
6500:
6485:
6481:
6463:
6459:
6451:
6449:
6446:
6445:
6439:
6423:
6403:
6399:
6388:
6385:
6384:
6380:
6342:
6339:
6338:
6330:
6272:
6270:
6268:
6265:
6264:
6253:
6178:
6176:
6173:
6172:
6161:
6158:
6146:
6138:
6129:
6123:
6113:
6110:
6101:
6095:
6092:antiderivatives
6084:
6078:
6074:
6071:
6061:
6054:
6050:
6046:
6012:
6007:
5994:
5963:
5958:
5945:
5933:
5930:
5929:
5926:
5917:
5911:
5907:
5873:
5868:
5855:
5824:
5819:
5806:
5781:
5776:
5763:
5744:
5739:
5729:
5725:
5704:
5699:
5689:
5685:
5670:
5666:
5664:
5661:
5660:
5649:
5626:
5621:
5611:
5607:
5586:
5581:
5571:
5567:
5552:
5548:
5546:
5543:
5542:
5520:
5519:
5498:
5493:
5480:
5449:
5444:
5431:
5406:
5401:
5388:
5377:
5371:
5370:
5359:
5358:
5349:
5336:
5314:
5301:
5285:
5272:
5261:
5255:
5254:
5248:
5244:
5235:
5216:
5212:
5203:
5190:
5186:
5177:
5166:
5159:
5157:
5154:
5153:
5149:
5146:
5137:
5131:
5128:
5119:
5113:
5109:
5100:
5093:
5073:
5069:
5063:
5059:
5044:
5040:
5034:
5030:
5022:
5019:
5018:
4992:
4988:
4977:
4965:
4961:
4934:
4930:
4924:
4920:
4905:
4901:
4899:
4896:
4895:
4877:
4865:
4861:
4850:
4839:
4830:
4826:
4818:
4814:
4803:
4799:
4795:
4791:
4788:
4779:
4773:
4726:
4722:
4702:
4690:
4686:
4650:
4646:
4640:
4636:
4612:
4608:
4606:
4603:
4602:
4598:
4595:
4586:
4579:
4573:
4570:
4564:
4561:
4555:
4552:
4546:
4542:
4539:
4529:
4526:
4516:
4505:
4461:
4457:
4430:
4426:
4414:
4410:
4408:
4405:
4404:
4401:Euler's formula
4362:
4358:
4352:
4348:
4321:
4317:
4311:
4307:
4305:
4302:
4301:
4291:
4280:
4254:
4244:
4240:
4228:
4224:
4215:
4211:
4206:
4203:
4202:
4194:
4187:
4162:
4158:
4152:
4148:
4136:
4132:
4126:
4122:
4120:
4117:
4116:
4112:
4108:
4101:
4094:
4088:
4085:
4079:
4065:
4057:
4053:
4016:
4012:
4010:
4007:
4006:
3969:
3955:
3953:
3950:
3949:
3946:
3903:
3899:
3893:
3889:
3887:
3884:
3883:
3845:
3841:
3835:
3831:
3829:
3826:
3825:
3790:
3786:
3780:
3776:
3774:
3771:
3770:
3735:
3731:
3725:
3721:
3719:
3716:
3715:
3712:Euler's formula
3701:
3691:
3649:
3644:
3642:
3639:
3638:
3637:
3636:application of
3630:
3605:
3601:
3589:
3585:
3565:
3561:
3555:
3551:
3550:
3546:
3525:
3520:
3519:
3515:
3513:
3510:
3509:
3501:
3472:
3467:
3465:
3462:
3461:
3460:
3456:
3437:
3433:
3429:
3419:
3415:
3411:
3407:
3384:
3380:
3374:
3370:
3368:
3365:
3364:
3330:
3326:
3313:
3309:
3281:
3278:
3277:
3257:
3253:
3240:
3236:
3220:
3216:
3203:
3199:
3197:
3194:
3193:
3189:
3182:
3178:
3140:
3136:
3124:
3120:
3108:
3104:
3102:
3099:
3098:
3067:
3053:
3039:
3025:
3023:
3020:
3019:
2989:
2977:
2948:
2944:
2938:
2934:
2919:
2915:
2909:
2905:
2893:
2889:
2880:
2876:
2874:
2871:
2870:
2850:
2846:
2840:
2836:
2821:
2817:
2811:
2807:
2795:
2791:
2782:
2778:
2776:
2773:
2772:
2765:
2759:
2733:
2729:
2723:
2719:
2713:
2709:
2691:
2687:
2681:
2677:
2671:
2667:
2655:
2651:
2642:
2638:
2626:
2622:
2616:
2612:
2610:
2607:
2606:
2602:
2596:
2590:
2581:
2575:
2546:
2542:
2536:
2532:
2515:
2509:
2505:
2494:
2488:
2484:
2472:
2468:
2466:
2463:
2462:
2450:
2444:
2440:
2433:
2423:
2417:
2404:
2395:
2389:
2360:
2356:
2350:
2346:
2329:
2323:
2319:
2308:
2302:
2298:
2286:
2282:
2280:
2277:
2276:
2269:
2260:
2256:
2243:
2235:
2233:
2229:
2225:
2222:
2213:
2203:
2199:
2196:
2187:
2181:
2152:
2148:
2142:
2138:
2114:
2110:
2104:
2100:
2082:
2078:
2076:
2073:
2072:
2053:
2049:
2045:
2037:
2027:
2002:
1983:
1979:
1940:
1937:
1936:
1898:
1894:
1879:
1875:
1858:
1843:
1839:
1828:
1813:
1809:
1788:
1784:
1782:
1779:
1778:
1755:
1751:
1747:
1741:
1737:
1735:
1720:
1716:
1697:
1692:
1677:
1673:
1655:
1651:
1643:
1640:
1639:
1625:complex numbers
1606:linear operator
1594:
1588:
1584:
1580:
1576:
1561:
1553:
1545:
1537:
1524:
1518:
1495:
1491:
1487:
1481:
1477:
1475:
1460:
1456:
1437:
1432:
1417:
1413:
1395:
1391:
1389:
1386:
1385:
1374:linear operator
1362:
1340:
1336:
1335:
1330:
1312:
1308:
1307:
1302:
1294:
1286:
1282:
1267:
1263:
1262:
1258:
1256:
1254:
1251:
1250:
1245:in the case of
1226:
1222:
1218:
1212:
1208:
1206:
1204:
1201:
1200:
1196:
1186:
1177:
1170:
1164:
1149:
1148:
1130:
1129:
1095:
1084:
1068:antiderivatives
1020:differentiation
993:
965:
961:
951:
936:
927:
919:
911:
905:
904:
866:
862:
847:
843:
829:
814:
810:
799:
784:
780:
759:
755:
753:
750:
749:
722:
693:
692:
691:
622:Jacob Bernoulli
606:
593:
592:
574:
543:Petrov–Galerkin
511:
496:
483:
475:
474:
473:
455:
401:Boundary values
390:
382:
381:
357:
344:
343:
342:
316:
310:
302:
301:
289:
266:
224:
180:
167:
166:
162:
140:Social sciences
96:
74:
55:
35:
28:
23:
22:
15:
12:
11:
5:
9891:
9881:
9880:
9863:
9862:
9860:
9859:
9854:
9849:
9844:
9839:
9834:
9829:
9824:
9819:
9817:Ernst Lindelöf
9814:
9809:
9804:
9799:
9797:Leonhard Euler
9794:
9789:
9783:
9781:
9780:Mathematicians
9777:
9776:
9774:
9773:
9768:
9763:
9758:
9752:
9750:
9746:
9745:
9742:
9741:
9739:
9738:
9733:
9728:
9723:
9718:
9713:
9708:
9703:
9698:
9693:
9688:
9683:
9678:
9673:
9668:
9662:
9660:
9656:
9655:
9653:
9652:
9647:
9642:
9636:
9631:
9626:
9621:
9616:
9611:
9606:
9604:Phase portrait
9601:
9595:
9593:
9589:
9588:
9586:
9585:
9580:
9575:
9570:
9564:
9562:
9555:
9551:
9550:
9547:
9546:
9544:
9543:
9538:
9537:
9536:
9526:
9519:
9517:
9513:
9512:
9510:
9509:
9507:On jet bundles
9504:
9499:
9494:
9489:
9484:
9479:
9474:
9472:Nonhomogeneous
9469:
9464:
9458:
9456:
9452:
9451:
9449:
9448:
9443:
9438:
9433:
9428:
9423:
9418:
9413:
9408:
9403:
9397:
9395:
9388:
9387:Classification
9384:
9383:
9376:
9375:
9368:
9361:
9353:
9347:
9346:
9340:
9333:
9332:External links
9330:
9329:
9328:
9323:
9310:
9305:
9292:
9287:
9271:
9270:
9255:
9235:
9148:
9138:
9137:
9135:
9132:
9131:
9130:
9125:
9120:
9115:
9110:
9103:
9100:
8949:Main article:
8946:
8943:
8926:
8923:
8920:
8916:
8912:
8909:
8906:
8901:
8897:
8874:
8871:
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8801:
8798:
8795:
8791:
8785:
8782:
8779:
8775:
8771:
8768:
8765:
8762:
8757:
8754:
8751:
8747:
8741:
8737:
8721:
8718:
8684:Ernest Vessiot
8663:
8660:
8648:
8645:
8642:
8638:
8635:
8632:
8628:
8624:
8621:
8618:
8613:
8610:
8606:
8600:
8593:
8589:
8584:
8580:
8577:
8574:
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8515:
8512:
8509:
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8499:
8478:
8473:
8468:
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8412:
8408:
8386:
8383:
8380:
8376:
8373:
8370:
8366:
8362:
8359:
8356:
8351:
8348:
8344:
8340:
8337:
8334:
8331:
8328:
8325:
8319:
8315:
8310:
8307:
8304:
8301:
8298:
8295:
8292:
8289:
8285:
8249:
8246:
8243:
8240:
8237:
8234:
8231:
8228:
8225:
8222:
8219:
8216:
8213:
8207:
8204:
8200:
8163:
8160:
8157:
8154:
8151:
8148:
8135:antiderivative
8101:
8098:
8095:
8092:
8058:
8054:
8050:
8047:
8043:
8039:
8012:
8008:
8004:
8000:
7996:
7993:
7989:
7985:
7947:
7944:
7941:
7937:
7910:
7906:
7879:
7874:
7870:
7866:
7863:
7860:
7855:
7852:
7849:
7845:
7841:
7838:
7835:
7830:
7826:
7822:
7819:
7816:
7811:
7808:
7805:
7801:
7797:
7794:
7791:
7788:
7783:
7779:
7775:
7772:
7770:
7768:
7765:
7762:
7758:
7754:
7750:
7746:
7745:
7742:
7737:
7735:
7733:
7728:
7724:
7720:
7717:
7714:
7709:
7706:
7703:
7699:
7695:
7692:
7689:
7684:
7680:
7676:
7673:
7670:
7665:
7662:
7659:
7655:
7651:
7648:
7645:
7642:
7637:
7633:
7629:
7626:
7624:
7622:
7619:
7616:
7612:
7608:
7604:
7600:
7599:
7551:
7546:
7543:
7540:
7536:
7532:
7528:
7524:
7520:
7493:
7489:
7485:
7481:
7478:
7451:
7447:
7443:
7440:
7437:
7432:
7428:
7401:
7398:
7395:
7391:
7387:
7384:
7381:
7377:
7374:
7370:
7366:
7363:
7334:Main article:
7331:
7328:
7316:
7311:
7307:
7304:
7301:
7295:
7290:
7286:
7282:
7279:
7276:
7273:
7270:
7250:
7247:
7244:
7241:
7238:
7235:
7232:
7212:
7209:
7205:
7201:
7198:
7193:
7189:
7185:
7182:
7179:
7176:
7173:
7153:
7150:
7147:
7142:
7138:
7134:
7131:
7128:
7109:
7104:
7100:
7096:
7093:
7089:
7086:
7082:
7079:
7076:
7056:
7051:
7047:
7043:
7040:
7037:
7034:
7030:
7027:
7023:
7001:
6996:
6993:
6988:
6985:
6965:
6960:
6957:
6952:
6949:
6944:
6940:
6937:
6914:
6911:
6906:
6902:
6899:
6896:
6893:
6887:
6884:
6881:
6878:
6874:
6871:
6850:
6847:
6844:
6841:
6836:
6832:
6829:
6826:
6823:
6817:
6814:
6811:
6808:
6804:
6801:
6788:
6785:
6760:
6757:
6754:
6749:
6746:
6742:
6738:
6735:
6730:
6726:
6722:
6717:
6713:
6709:
6706:
6703:
6683:
6678:
6675:
6671:
6667:
6664:
6660:
6654:
6651:
6647:
6643:
6639:
6632:
6629:
6625:
6597:
6593:
6588:
6585:
6581:
6577:
6569:
6566:
6562:
6556:
6551:
6548:
6544:
6540:
6537:
6515:
6510:
6507:
6503:
6499:
6496:
6491:
6488:
6484:
6480:
6477:
6474:
6469:
6466:
6462:
6457:
6454:
6411:
6406:
6402:
6398:
6395:
6392:
6377:antiderivative
6363:
6360:
6356:
6353:
6349:
6346:
6318:
6315:
6312:
6309:
6306:
6303:
6300:
6297:
6293:
6290:
6287:
6282:
6278:
6275:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6215:
6212:
6209:
6206:
6203:
6200:
6197:
6194:
6191:
6188:
6184:
6181:
6157:
6154:
6142:
6134:
6127:
6121:
6106:
6099:
6088:linear algebra
6082:
6067:
6059:
6032:
6027:
6024:
6021:
6018:
6015:
6010:
6006:
6001:
5997:
5993:
5989:
5986:
5983:
5978:
5975:
5972:
5969:
5966:
5961:
5957:
5952:
5948:
5944:
5940:
5937:
5922:
5915:
5893:
5888:
5885:
5882:
5879:
5876:
5871:
5867:
5862:
5858:
5854:
5850:
5847:
5844:
5839:
5836:
5833:
5830:
5827:
5822:
5818:
5813:
5809:
5805:
5801:
5796:
5793:
5790:
5787:
5784:
5779:
5775:
5770:
5766:
5762:
5758:
5753:
5750:
5747:
5742:
5738:
5732:
5728:
5724:
5721:
5718:
5713:
5710:
5707:
5702:
5698:
5692:
5688:
5684:
5679:
5676:
5673:
5669:
5635:
5632:
5629:
5624:
5620:
5614:
5610:
5606:
5603:
5600:
5595:
5592:
5589:
5584:
5580:
5574:
5570:
5566:
5561:
5558:
5555:
5551:
5518:
5513:
5510:
5507:
5504:
5501:
5496:
5492:
5487:
5483:
5479:
5475:
5472:
5469:
5464:
5461:
5458:
5455:
5452:
5447:
5443:
5438:
5434:
5430:
5426:
5421:
5418:
5415:
5412:
5409:
5404:
5400:
5395:
5391:
5387:
5383:
5380:
5378:
5376:
5373:
5372:
5369:
5364:
5362:
5360:
5356:
5352:
5348:
5343:
5339:
5335:
5331:
5328:
5325:
5321:
5317:
5313:
5308:
5304:
5300:
5296:
5292:
5288:
5284:
5279:
5275:
5271:
5267:
5264:
5262:
5260:
5257:
5256:
5251:
5247:
5242:
5238:
5234:
5230:
5227:
5224:
5219:
5215:
5210:
5206:
5202:
5198:
5193:
5189:
5184:
5180:
5176:
5172:
5169:
5167:
5165:
5162:
5161:
5142:
5135:
5124:
5117:
5105:
5098:
5081:
5076:
5072:
5066:
5062:
5058:
5055:
5052:
5047:
5043:
5037:
5033:
5029:
5026:
5006:
5003:
5000:
4995:
4991:
4987:
4983:
4980:
4974:
4971:
4968:
4964:
4960:
4957:
4954:
4949:
4946:
4943:
4940:
4937:
4933:
4927:
4923:
4919:
4914:
4911:
4908:
4904:
4784:
4777:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4729:
4725:
4721:
4718:
4715:
4712:
4708:
4705:
4699:
4696:
4693:
4689:
4685:
4682:
4679:
4676:
4673:
4670:
4665:
4662:
4659:
4656:
4653:
4649:
4643:
4639:
4635:
4632:
4629:
4626:
4621:
4618:
4615:
4611:
4594:
4591:
4583:Cauchy problem
4577:
4568:
4559:
4550:
4537:
4524:
4502:
4501:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4464:
4460:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4433:
4429:
4425:
4420:
4417:
4413:
4388:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4361:
4355:
4351:
4347:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4320:
4314:
4310:
4277:
4266:
4261:
4257:
4253:
4250:
4247:
4243:
4239:
4236:
4231:
4227:
4223:
4218:
4214:
4210:
4184:
4173:
4168:
4165:
4161:
4155:
4151:
4147:
4142:
4139:
4135:
4129:
4125:
4092:
4083:
4039:
4036:
4033:
4030:
4027:
4024:
4019:
4015:
3994:
3991:
3988:
3985:
3982:
3979:
3975:
3972:
3968:
3965:
3961:
3958:
3945:
3942:
3929:
3926:
3923:
3920:
3917:
3914:
3909:
3906:
3902:
3896:
3892:
3871:
3868:
3865:
3862:
3859:
3856:
3851:
3848:
3844:
3838:
3834:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3789:
3783:
3779:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3734:
3728:
3724:
3664:
3661:
3655:
3652:
3648:
3616:
3611:
3608:
3604:
3598:
3595:
3592:
3588:
3584:
3581:
3577:
3571:
3568:
3564:
3558:
3554:
3549:
3544:
3540:
3537:
3531:
3528:
3524:
3518:
3487:
3484:
3478:
3475:
3471:
3395:
3390:
3387:
3383:
3377:
3373:
3361:multiple roots
3351:
3350:
3338:
3333:
3329:
3325:
3321:
3316:
3312:
3307:
3304:
3301:
3298:
3294:
3291:
3288:
3285:
3265:
3260:
3256:
3252:
3248:
3243:
3239:
3234:
3229:
3226:
3223:
3219:
3214:
3209:
3206:
3202:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3143:
3139:
3135:
3132:
3127:
3123:
3119:
3116:
3111:
3107:
3086:
3083:
3080:
3077:
3073:
3070:
3066:
3063:
3059:
3056:
3052:
3049:
3045:
3042:
3038:
3035:
3031:
3028:
3015:
3014:
2959:
2956:
2951:
2947:
2941:
2937:
2933:
2930:
2927:
2922:
2918:
2912:
2908:
2904:
2901:
2896:
2892:
2888:
2883:
2879:
2853:
2849:
2843:
2839:
2835:
2832:
2829:
2824:
2820:
2814:
2810:
2806:
2803:
2798:
2794:
2790:
2785:
2781:
2758:Factoring out
2747:
2744:
2739:
2736:
2732:
2726:
2722:
2716:
2712:
2708:
2705:
2702:
2697:
2694:
2690:
2684:
2680:
2674:
2670:
2666:
2661:
2658:
2654:
2650:
2645:
2641:
2637:
2632:
2629:
2625:
2619:
2615:
2586:
2579:
2563:
2560:
2555:
2552:
2549:
2545:
2539:
2535:
2531:
2528:
2525:
2521:
2518:
2512:
2508:
2504:
2500:
2497:
2491:
2487:
2483:
2480:
2475:
2471:
2411:Leonhard Euler
2400:
2393:
2377:
2374:
2369:
2366:
2363:
2359:
2353:
2349:
2345:
2342:
2339:
2335:
2332:
2326:
2322:
2318:
2314:
2311:
2305:
2301:
2297:
2294:
2289:
2285:
2268:
2265:
2239:
2218:
2211:
2192:
2185:
2169:
2166:
2163:
2160:
2155:
2151:
2145:
2141:
2137:
2134:
2131:
2128:
2125:
2122:
2117:
2113:
2107:
2103:
2099:
2096:
2093:
2090:
2085:
2081:
2071:have the form
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1924:
1921:
1918:
1915:
1912:
1907:
1904:
1901:
1897:
1893:
1890:
1887:
1882:
1878:
1874:
1871:
1868:
1864:
1861:
1857:
1854:
1851:
1846:
1842:
1838:
1834:
1831:
1827:
1824:
1821:
1816:
1812:
1808:
1805:
1802:
1799:
1796:
1791:
1787:
1766:
1758:
1754:
1750:
1744:
1740:
1734:
1731:
1728:
1723:
1719:
1715:
1712:
1709:
1703:
1700:
1696:
1691:
1688:
1685:
1680:
1676:
1672:
1669:
1666:
1663:
1658:
1654:
1650:
1647:
1583:to a function
1557:
1533:
1522:
1506:
1498:
1494:
1490:
1484:
1480:
1474:
1471:
1468:
1463:
1459:
1455:
1452:
1449:
1443:
1440:
1436:
1431:
1428:
1425:
1420:
1416:
1412:
1409:
1406:
1403:
1398:
1394:
1343:
1339:
1333:
1329:
1325:
1322:
1315:
1311:
1305:
1301:
1297:
1289:
1285:
1281:
1278:
1275:
1270:
1266:
1261:
1229:
1225:
1221:
1215:
1211:
1166:Main article:
1163:
1160:
1083:
1080:
1052:error function
992:
989:
946:are arbitrary
923:
909:
892:
889:
886:
883:
880:
875:
872:
869:
865:
861:
858:
855:
850:
846:
842:
839:
835:
832:
828:
825:
822:
817:
813:
809:
805:
802:
798:
795:
792:
787:
783:
779:
776:
773:
770:
767:
762:
758:
724:
723:
721:
720:
713:
706:
698:
695:
694:
690:
689:
684:
679:
674:
672:Ernst Lindelöf
669:
664:
659:
654:
649:
644:
642:Joseph Fourier
639:
634:
629:
627:Leonhard Euler
624:
619:
614:
608:
607:
604:
603:
600:
599:
595:
594:
591:
590:
585:
580:
573:
572:
567:
562:
557:
552:
547:
546:
545:
535:
530:
529:
528:
521:Finite element
518:
514:Crank–Nicolson
505:
500:
494:
489:
485:
484:
481:
480:
477:
476:
472:
471:
466:
461:
453:
448:
435:
433:Phase portrait
430:
425:
424:
423:
421:Cauchy problem
418:
413:
408:
398:
392:
391:
389:General topics
388:
387:
384:
383:
380:
379:
374:
369:
364:
358:
355:
354:
351:
350:
346:
345:
341:
340:
335:
334:
333:
322:
321:
320:
311:
308:
307:
304:
303:
298:
297:
296:
295:
288:
287:
282:
276:
273:
272:
268:
267:
265:
264:
262:Nonhomogeneous
255:
250:
247:
241:
240:
239:
231:
230:
226:
225:
223:
222:
217:
212:
207:
202:
197:
192:
186:
181:
178:
177:
174:
173:
172:Classification
169:
168:
159:
158:
157:
156:
151:
143:
142:
136:
135:
134:
133:
128:
123:
115:
114:
108:
107:
106:
105:
100:
94:
89:
84:
76:
75:
73:
72:
67:
61:
56:
53:
52:
49:
48:
44:
43:
26:
9:
6:
4:
3:
2:
9890:
9879:
9876:
9875:
9873:
9858:
9855:
9853:
9850:
9848:
9845:
9843:
9840:
9838:
9835:
9833:
9830:
9828:
9825:
9823:
9820:
9818:
9815:
9813:
9810:
9808:
9805:
9803:
9800:
9798:
9795:
9793:
9790:
9788:
9785:
9784:
9782:
9778:
9772:
9769:
9767:
9764:
9762:
9759:
9757:
9754:
9753:
9751:
9747:
9737:
9734:
9732:
9729:
9727:
9724:
9722:
9719:
9717:
9714:
9712:
9709:
9707:
9704:
9702:
9699:
9697:
9694:
9692:
9689:
9687:
9684:
9682:
9679:
9677:
9674:
9672:
9669:
9667:
9664:
9663:
9661:
9657:
9651:
9648:
9646:
9643:
9640:
9637:
9635:
9632:
9630:
9627:
9625:
9622:
9620:
9617:
9615:
9612:
9610:
9607:
9605:
9602:
9600:
9597:
9596:
9594:
9590:
9584:
9581:
9579:
9576:
9574:
9571:
9569:
9566:
9565:
9563:
9559:
9556:
9552:
9542:
9539:
9535:
9532:
9531:
9530:
9527:
9524:
9521:
9520:
9518:
9514:
9508:
9505:
9503:
9500:
9498:
9495:
9493:
9490:
9488:
9485:
9483:
9480:
9478:
9475:
9473:
9470:
9468:
9465:
9463:
9460:
9459:
9457:
9453:
9447:
9444:
9442:
9439:
9437:
9434:
9432:
9429:
9427:
9424:
9422:
9419:
9417:
9414:
9412:
9409:
9407:
9404:
9402:
9399:
9398:
9396:
9392:
9389:
9385:
9381:
9374:
9369:
9367:
9362:
9360:
9355:
9354:
9351:
9344:
9341:
9339:
9336:
9335:
9326:
9324:0-521-82650-0
9320:
9316:
9311:
9308:
9302:
9298:
9293:
9290:
9288:0-471-07411-X
9284:
9280:
9275:
9274:
9267:
9266:
9259:
9252:
9251:
9244:
9242:
9240:
9231:
9227:
9223:
9217:
9213:
9209:. This means
9206:
9202:
9198:
9192:
9185:
9181:
9177:
9171:
9167:
9163:
9158:
9152:
9143:
9139:
9129:
9126:
9124:
9121:
9119:
9116:
9114:
9111:
9109:
9106:
9105:
9099:
9097:
9093:
9092:singularities
9089:
9085:
9081:
9077:
9073:
9068:
9066:
9062:
9058:
9054:
9053:Taylor series
9050:
9046:
9041:
9038:
9036:
9032:
9028:
9024:
9019:
9017:
9013:
9009:
9005:
9001:
8997:
8993:
8989:
8985:
8981:
8977:
8973:
8969:
8964:
8962:
8958:
8952:
8942:
8924:
8921:
8918:
8914:
8910:
8907:
8904:
8899:
8895:
8872:
8869:
8866:
8860:
8854:
8849:
8845:
8841:
8838:
8835:
8829:
8818:
8815:
8812:
8805:
8799:
8796:
8793:
8789:
8783:
8780:
8777:
8773:
8769:
8763:
8752:
8745:
8739:
8735:
8726:
8717:
8715:
8710:
8706:
8704:
8700:
8696:
8691:
8689:
8685:
8681:
8677:
8673:
8669:
8659:
8646:
8643:
8640:
8633:
8619:
8611:
8608:
8604:
8598:
8591:
8587:
8582:
8575:
8569:
8566:
8546:
8542:
8533:
8530:
8526:
8519:
8513:
8510:
8504:
8476:
8471:
8461:
8453:
8449:
8431:
8429:
8384:
8381:
8378:
8371:
8357:
8349:
8346:
8342:
8338:
8332:
8326:
8323:
8305:
8299:
8296:
8290:
8269:
8267:
8263:
8247:
8241:
8235:
8232:
8229:
8226:
8220:
8214:
8211:
8205:
8202:
8198:
8184:
8161:
8158:
8155:
8152:
8149:
8146:
8136:
8122:
8117:
8096:
8090:
8081:of functions
8080:
8079:square matrix
8073:of dimension
8072:
8056:
8048:
8045:
8041:
8026:
8023:
8010:
8002:
7994:
7991:
7987:
7971:
7945:
7942:
7939:
7935:
7908:
7904:
7877:
7872:
7868:
7861:
7853:
7850:
7847:
7843:
7839:
7836:
7833:
7828:
7824:
7817:
7809:
7806:
7803:
7799:
7795:
7789:
7781:
7777:
7773:
7771:
7763:
7756:
7752:
7748:
7740:
7736:
7726:
7722:
7715:
7707:
7704:
7701:
7697:
7693:
7690:
7687:
7682:
7678:
7671:
7663:
7660:
7657:
7653:
7649:
7643:
7635:
7631:
7627:
7625:
7617:
7610:
7606:
7602:
7589:
7576:
7572:
7568:
7549:
7544:
7541:
7538:
7534:
7530:
7526:
7522:
7518:
7491:
7487:
7483:
7479:
7476:
7449:
7445:
7441:
7438:
7435:
7430:
7426:
7396:
7389:
7385:
7382:
7379:
7375:
7372:
7368:
7364:
7361:
7348:
7343:
7337:
7327:
7314:
7309:
7305:
7302:
7299:
7293:
7288:
7284:
7280:
7274:
7268:
7248:
7245:
7242:
7236:
7230:
7210:
7207:
7203:
7199:
7196:
7191:
7187:
7183:
7177:
7171:
7151:
7148:
7145:
7140:
7136:
7132:
7129:
7126:
7107:
7102:
7098:
7094:
7091:
7087:
7080:
7077:
7054:
7049:
7045:
7041:
7038:
7035:
7032:
7028:
7025:
7021:
7012:
6999:
6994:
6991:
6986:
6983:
6963:
6958:
6955:
6950:
6947:
6942:
6938:
6935:
6912:
6909:
6904:
6897:
6891:
6885:
6879:
6872:
6869:
6848:
6845:
6842:
6839:
6834:
6827:
6821:
6815:
6809:
6802:
6799:
6784:
6758:
6755:
6752:
6747:
6744:
6740:
6736:
6733:
6728:
6724:
6720:
6715:
6711:
6707:
6704:
6701:
6681:
6676:
6673:
6669:
6665:
6662:
6658:
6652:
6649:
6645:
6641:
6637:
6630:
6627:
6623:
6613:
6595:
6591:
6586:
6583:
6579:
6575:
6567:
6564:
6560:
6554:
6549:
6546:
6542:
6538:
6535:
6513:
6508:
6505:
6501:
6497:
6494:
6489:
6486:
6482:
6478:
6475:
6472:
6467:
6464:
6460:
6455:
6452:
6442:
6438:
6433:
6430:
6426:
6409:
6404:
6400:
6396:
6393:
6390:
6378:
6361:
6358:
6354:
6351:
6347:
6344:
6336:
6316:
6313:
6310:
6307:
6304:
6301:
6298:
6295:
6291:
6288:
6285:
6280:
6276:
6273:
6260:
6256:
6250:
6237:
6231:
6225:
6222:
6216:
6210:
6204:
6198:
6195:
6189:
6182:
6179:
6168:
6164:
6153:
6149:
6145:
6141:
6137:
6133:
6126:
6120:
6116:
6109:
6105:
6098:
6093:
6089:
6081:
6070:
6065:
6057:
6043:
6030:
6022:
6019:
6016:
6008:
6004:
5999:
5995:
5991:
5987:
5984:
5981:
5973:
5970:
5967:
5959:
5955:
5950:
5946:
5942:
5938:
5935:
5925:
5921:
5914:
5904:
5891:
5883:
5880:
5877:
5869:
5865:
5860:
5856:
5852:
5848:
5845:
5842:
5834:
5831:
5828:
5820:
5816:
5811:
5807:
5803:
5799:
5791:
5788:
5785:
5777:
5773:
5768:
5764:
5760:
5756:
5748:
5740:
5736:
5730:
5726:
5722:
5719:
5716:
5708:
5700:
5696:
5690:
5686:
5682:
5674:
5667:
5656:
5652:
5630:
5622:
5618:
5612:
5608:
5604:
5601:
5598:
5590:
5582:
5578:
5572:
5568:
5564:
5556:
5549:
5540:
5536:
5516:
5508:
5505:
5502:
5494:
5490:
5485:
5481:
5477:
5473:
5470:
5467:
5459:
5456:
5453:
5445:
5441:
5436:
5432:
5428:
5424:
5416:
5413:
5410:
5402:
5398:
5393:
5389:
5385:
5381:
5379:
5374:
5367:
5363:
5354:
5350:
5346:
5341:
5337:
5333:
5329:
5326:
5323:
5319:
5315:
5311:
5306:
5302:
5298:
5294:
5290:
5286:
5282:
5277:
5273:
5269:
5265:
5263:
5258:
5249:
5245:
5240:
5236:
5232:
5228:
5225:
5222:
5217:
5213:
5208:
5204:
5200:
5196:
5191:
5187:
5182:
5178:
5174:
5170:
5168:
5163:
5145:
5141:
5134:
5127:
5123:
5116:
5108:
5104:
5097:
5079:
5074:
5070:
5064:
5060:
5056:
5053:
5050:
5045:
5041:
5035:
5031:
5027:
5024:
5004:
5001:
4998:
4993:
4989:
4985:
4981:
4978:
4972:
4969:
4966:
4962:
4958:
4955:
4952:
4944:
4941:
4938:
4931:
4925:
4921:
4917:
4909:
4902:
4892:
4890:
4885:
4883:
4876:applies when
4875:
4871:
4857:
4853:
4846:
4842:
4836:
4833:
4824:
4811:
4807:
4787:
4783:
4776:
4759:
4753:
4747:
4744:
4738:
4732:
4727:
4723:
4719:
4713:
4706:
4703:
4697:
4694:
4691:
4687:
4683:
4680:
4677:
4671:
4660:
4657:
4654:
4647:
4641:
4637:
4633:
4627:
4616:
4609:
4590:
4584:
4576:
4567:
4558:
4549:
4536:
4532:
4523:
4519:
4512:
4508:
4488:
4479:
4476:
4470:
4467:
4462:
4458:
4454:
4448:
4445:
4439:
4436:
4431:
4427:
4418:
4415:
4411:
4402:
4386:
4381:
4375:
4372:
4369:
4366:
4359:
4353:
4349:
4345:
4340:
4334:
4331:
4328:
4325:
4318:
4312:
4308:
4298:
4294:
4289:
4283:
4278:
4264:
4259:
4255:
4251:
4248:
4245:
4241:
4234:
4229:
4225:
4221:
4216:
4212:
4198:
4190:
4185:
4171:
4166:
4163:
4159:
4153:
4149:
4145:
4140:
4137:
4133:
4127:
4123:
4104:
4099:
4098:
4097:
4091:
4082:
4076:
4072:
4068:
4063:
4050:
4037:
4034:
4031:
4028:
4025:
4022:
4017:
4013:
3992:
3989:
3986:
3983:
3980:
3977:
3973:
3970:
3966:
3963:
3959:
3956:
3941:
3924:
3921:
3915:
3912:
3907:
3904:
3900:
3894:
3890:
3866:
3863:
3857:
3854:
3849:
3846:
3842:
3836:
3832:
3809:
3803:
3800:
3797:
3794:
3787:
3781:
3777:
3754:
3748:
3745:
3742:
3739:
3732:
3726:
3722:
3713:
3708:
3704:
3698:
3694:
3689:
3684:
3682:
3677:
3662:
3659:
3653:
3650:
3646:
3633:
3627:
3614:
3609:
3606:
3602:
3596:
3593:
3590:
3586:
3582:
3579:
3575:
3569:
3566:
3562:
3556:
3552:
3547:
3542:
3538:
3535:
3529:
3526:
3522:
3516:
3507:
3485:
3482:
3476:
3473:
3469:
3452:
3448:
3444:
3440:
3426:
3422:
3393:
3388:
3385:
3381:
3375:
3371:
3362:
3358:
3349:
3336:
3331:
3327:
3323:
3319:
3314:
3310:
3305:
3302:
3299:
3296:
3292:
3289:
3286:
3283:
3263:
3258:
3254:
3250:
3246:
3241:
3237:
3232:
3227:
3224:
3221:
3217:
3212:
3207:
3204:
3200:
3186:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3141:
3137:
3133:
3130:
3125:
3121:
3117:
3114:
3109:
3105:
3084:
3081:
3078:
3075:
3071:
3068:
3064:
3061:
3057:
3054:
3050:
3047:
3043:
3040:
3036:
3033:
3029:
3026:
3017:
3016:
3012:
3011:
3008:
3006:
3002:
2996:
2992:
2987:
2983:
2975:
2970:
2957:
2954:
2949:
2945:
2939:
2935:
2931:
2928:
2925:
2920:
2916:
2910:
2906:
2902:
2899:
2894:
2890:
2886:
2881:
2877:
2869:
2851:
2847:
2841:
2837:
2833:
2830:
2827:
2822:
2818:
2812:
2808:
2804:
2801:
2796:
2792:
2788:
2783:
2779:
2771:
2762:
2745:
2742:
2737:
2734:
2730:
2724:
2720:
2714:
2710:
2706:
2703:
2700:
2695:
2692:
2688:
2682:
2678:
2672:
2668:
2664:
2659:
2656:
2652:
2648:
2643:
2639:
2635:
2630:
2627:
2623:
2617:
2613:
2599:
2593:
2589:
2585:
2578:
2561:
2558:
2550:
2543:
2537:
2533:
2529:
2526:
2523:
2519:
2516:
2510:
2506:
2502:
2498:
2495:
2489:
2485:
2481:
2478:
2473:
2469:
2459:
2456:
2453:
2447:
2436:
2430:
2426:
2420:
2416:
2412:
2407:
2403:
2399:
2392:
2375:
2372:
2364:
2357:
2351:
2347:
2343:
2340:
2337:
2333:
2330:
2324:
2320:
2316:
2312:
2309:
2303:
2299:
2295:
2292:
2287:
2283:
2274:
2264:
2253:
2247:
2242:
2238:
2221:
2217:
2210:
2206:
2195:
2191:
2184:
2167:
2161:
2153:
2149:
2143:
2139:
2135:
2132:
2129:
2123:
2115:
2111:
2105:
2101:
2097:
2091:
2083:
2079:
2068:
2064:
2060:
2056:
2043:
2034:
2030:
2025:
2021:
2017:
2012:
2009:
2005:
1998:
1994:
1990:
1986:
1976:
1963:
1957:
1951:
1948:
1945:
1942:
1919:
1913:
1910:
1902:
1895:
1888:
1880:
1876:
1872:
1869:
1866:
1862:
1859:
1852:
1844:
1840:
1836:
1832:
1829:
1822:
1814:
1810:
1806:
1803:
1797:
1789:
1785:
1764:
1756:
1752:
1748:
1742:
1738:
1729:
1721:
1717:
1713:
1710:
1707:
1701:
1698:
1694:
1686:
1678:
1674:
1670:
1664:
1656:
1652:
1648:
1645:
1636:
1634:
1630:
1626:
1622:
1618:
1613:
1611:
1607:
1601:
1597:
1591:
1573:
1571:
1570:zero function
1565:
1560:
1556:
1551:
1541:
1536:
1532:
1528:
1521:
1504:
1496:
1492:
1488:
1482:
1478:
1469:
1461:
1457:
1453:
1450:
1447:
1441:
1438:
1434:
1426:
1418:
1414:
1410:
1404:
1396:
1392:
1383:
1379:
1375:
1371:
1366:
1341:
1337:
1331:
1327:
1320:
1313:
1309:
1303:
1299:
1287:
1283:
1279:
1276:
1273:
1268:
1264:
1248:
1227:
1223:
1219:
1213:
1209:
1194:
1190:
1189:th derivative
1183:
1175:
1169:
1159:
1157:
1153:
1144:
1142:
1138:
1134:
1126:
1123:, as it is a
1122:
1121:
1116:
1115:zero function
1112:
1108:
1107:constant term
1102:
1098:
1093:
1089:
1079:
1077:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1025:
1021:
1017:
1013:
1008:
1006:
1002:
998:
988:
986:
982:
978:
974:
969:
958:
954:
949:
943:
939:
931:
926:
922:
915:
908:
887:
881:
878:
870:
863:
856:
848:
844:
840:
837:
833:
830:
823:
815:
811:
807:
803:
800:
793:
785:
781:
777:
774:
768:
760:
756:
748:of the form
747:
743:
739:
735:
731:
719:
714:
712:
707:
705:
700:
699:
697:
696:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
609:
602:
601:
597:
596:
589:
586:
584:
581:
579:
576:
575:
571:
568:
566:
563:
561:
558:
556:
553:
551:
548:
544:
541:
540:
539:
536:
534:
533:Finite volume
531:
527:
524:
523:
522:
519:
515:
509:
506:
504:
501:
499:
495:
493:
490:
487:
486:
479:
478:
470:
467:
465:
462:
458:
454:
452:
449:
447:
443:
439:
436:
434:
431:
429:
426:
422:
419:
417:
414:
412:
409:
407:
404:
403:
402:
399:
397:
394:
393:
386:
385:
378:
375:
373:
370:
368:
365:
363:
360:
359:
353:
352:
348:
347:
339:
336:
332:
329:
328:
327:
324:
323:
319:
313:
312:
306:
305:
294:
291:
290:
286:
283:
281:
278:
277:
275:
274:
270:
269:
263:
259:
256:
254:
251:
248:
246:
243:
242:
238:
235:
234:
233:
232:
228:
227:
221:
218:
216:
213:
211:
208:
206:
203:
201:
198:
196:
193:
191:
188:
187:
185:
184:
176:
175:
171:
170:
165:
155:
152:
150:
147:
146:
145:
144:
141:
138:
137:
132:
129:
127:
124:
122:
119:
118:
117:
116:
113:
110:
109:
104:
101:
99:
95:
93:
90:
88:
85:
83:
80:
79:
78:
77:
71:
68:
66:
63:
62:
60:
59:
51:
50:
46:
45:
42:
39:
38:
33:
19:
9852:Martin Kutta
9807:Émile Picard
9787:Isaac Newton
9701:Euler method
9671:Substitution
9435:
9314:
9296:
9278:
9263:
9258:
9248:
9229:
9225:
9221:
9215:
9211:
9204:
9200:
9196:
9190:
9183:
9179:
9175:
9169:
9165:
9161:
9151:
9142:
9069:
9064:
9057:power series
9044:
9042:
9039:
9020:
8965:
8960:
8954:
8723:
8711:
8707:
8692:
8680:Émile Picard
8665:
8432:
8270:
8120:
8071:vector space
8027:
8024:
7969:
7577:
7570:
7566:
7349:
7345:
7013:
6790:
6612:product rule
6440:
6434:
6428:
6424:
6258:
6254:
6251:
6166:
6162:
6159:
6150:
6143:
6139:
6135:
6131:
6124:
6118:
6114:
6107:
6103:
6096:
6079:
6068:
6063:
6055:
6044:
5923:
5919:
5912:
5905:
5654:
5650:
5535:product rule
5143:
5139:
5132:
5125:
5121:
5114:
5106:
5102:
5095:
4893:
4886:
4855:
4851:
4844:
4840:
4834:
4831:
4812:
4805:
4785:
4781:
4774:
4596:
4574:
4565:
4556:
4547:
4534:
4530:
4521:
4517:
4510:
4506:
4503:
4296:
4292:
4281:
4196:
4188:
4102:
4089:
4080:
4074:
4070:
4066:
4051:
3947:
3706:
3702:
3696:
3692:
3685:
3678:
3631:
3628:
3450:
3446:
3442:
3438:
3424:
3420:
3357:simple roots
3354:
3184:
3018:
3005:vector space
2994:
2990:
2971:
2769:
2760:
2597:
2594:
2587:
2583:
2576:
2460:
2454:
2451:
2445:
2434:
2428:
2424:
2418:
2408:
2401:
2397:
2390:
2272:
2270:
2251:
2250:| >
2245:
2240:
2236:
2219:
2215:
2208:
2204:
2193:
2189:
2182:
2066:
2062:
2058:
2054:
2035:
2028:
2024:vector space
2015:
2013:
2007:
2003:
1996:
1992:
1988:
1984:
1977:
1637:
1621:real numbers
1617:vector space
1614:
1599:
1595:
1589:
1574:
1563:
1558:
1554:
1549:
1539:
1534:
1530:
1526:
1519:
1377:
1376:or, simply,
1373:
1369:
1367:
1173:
1171:
1156:vector space
1147:
1145:
1136:
1128:
1118:
1106:
1100:
1096:
1091:
1086:The highest
1085:
1009:
994:
976:
970:
956:
952:
941:
937:
929:
924:
920:
913:
906:
733:
727:
677:Émile Picard
662:Martin Kutta
652:George Green
612:Isaac Newton
444: /
440: /
260: /
214:
126:Chaos theory
9609:Phase space
9467:Homogeneous
9006:, and many
8968:polynomials
8183:exponential
8116:determinant
6112:, and then
4515:satisfying
3679:As, by the
1629:free module
1568:is not the
1120:homogeneous
1024:integration
730:mathematics
570:Runge–Kutta
315:Difference
258:Homogeneous
70:Engineering
9837:John Crank
9666:Inspection
9529:Stochastic
9523:Difference
9497:Autonomous
9441:Non-linear
9431:Fractional
9394:Operations
9219:, so that
9134:References
9080:indefinite
9076:derivative
9065:vice versa
9035:algorithms
9027:derivative
8668:quadrature
8181:equal the
7569:= 1, ...,
7340:See also:
6437:reciprocal
5653:= 1, ...,
2993:= 0, ...,
2976:, one has
2605:such that
2432:such that
2255:for every
2232:such that
1247:univariate
1012:polynomial
997:quadrature
687:John Crank
488:Inspection
442:Asymptotic
326:Stochastic
245:Autonomous
220:Non-linear
210:Fractional
9641:solutions
9599:Wronskian
9554:Solutions
9482:Decoupled
9446:Holonomic
9194:, namely
9031:integrals
8976:logarithm
8922:−
8908:…
8839:⋯
8816:−
8797:−
8781:−
8672:integrals
8609:−
8583:∫
8531:−
8347:−
8339:∫
8236:
8215:
8153:∫
7837:⋯
7741:⋮
7691:⋯
7439:…
7383:…
7303:−
7300:α
7246:α
7067:That is
6976:that is
6951:−
6745:−
6734:∫
6674:−
6650:−
6584:−
6547:−
6536:−
6506:−
6487:−
6473:−
6465:−
6352:∫
6299:
6020:−
5985:⋯
5971:−
5881:−
5846:⋯
5832:−
5789:−
5720:⋯
5602:⋯
5539:induction
5506:−
5471:⋯
5457:−
5414:−
5368:⋮
5327:⋯
5226:⋯
5054:⋯
4970:−
4956:⋯
4942:−
4695:−
4681:⋯
4658:−
4477:β
4471:
4446:β
4440:
4416:α
4373:β
4370:−
4367:α
4332:β
4326:α
4246:−
4164:β
4138:α
3916:
3858:
3798:−
3663:α
3660:−
3607:α
3594:−
3567:α
3539:α
3536:−
3486:α
3483:−
3386:α
3300:
3287:
3222:−
3147:−
3115:−
3062:−
3034:−
2929:⋯
2831:⋯
2735:α
2721:α
2704:⋯
2693:α
2679:α
2657:α
2649:α
2628:α
2527:⋯
2341:⋯
2133:⋯
1870:⋯
1711:⋯
1631:over the
1619:over the
1451:⋯
1324:∂
1321:⋯
1296:∂
1277:⋯
1260:∂
1195:of order
1176:of order
1036:logarithm
1001:integrals
975:(ODE). A
838:⋯
428:Wronskian
406:Dirichlet
149:Economics
92:Chemistry
82:Astronomy
9872:Category
9749:Examples
9639:Integral
9411:Ordinary
9102:See also
9072:calculus
9010:such as
8114:, whose
8042:′
7988:′
7925:and the
7757:′
7611:′
7527:′
7480:′
7376:″
7365:′
7088:′
7029:′
6939:′
6873:′
6803:′
6456:′
6277:′
6183:′
6000:′
5951:′
5861:′
5812:′
5769:′
5486:′
5437:′
5394:′
5355:′
5342:′
5320:′
5307:′
5291:′
5278:′
5241:′
5209:′
5183:′
4982:′
4860:, where
4707:′
3974:′
3960:″
3072:′
3058:″
3044:‴
3030:⁗
3013:Example
2974:distinct
2520:″
2499:′
2334:″
2313:′
1863:″
1833:′
1529:), ...,
1378:operator
1150:solution
1139:if only
1064:calculus
1030:such as
955:′, ...,
834:″
804:′
746:equation
538:Galerkin
438:Lyapunov
349:Solution
293:Notation
285:Operator
271:Features
190:Ordinary
9477:Coupled
9416:Partial
8939:
8887:
8175:
8138:
8112:
8083:
7960:
7927:
7923:
7896:
7562:
7510:
7506:
7468:
7464:
7418:
7414:
7353:
6787:Example
6608:
6528:
6375:is any
6171:, is:
6102:, ...,
6062:, ...,
5918:, ...,
5138:, ...,
5120:, ...,
5101:, ...,
4780:, ...,
4533:′(0) =
3003:of the
2582:, ...,
2437:(0) = 1
2396:, ...,
2214:, ...,
2188:, ...,
1623:or the
1548:is the
1380:) is a
1184:to its
918:, ...,
411:Neumann
195:Partial
103:Geology
98:Biology
87:Physics
9492:Degree
9436:Linear
9321:
9303:
9285:
9088:limits
8988:cosine
8885:where
7894:where
6925:gives
6771:where
6422:where
6329:where
6130:+ ⋯ +
6094:gives
6077:, the
5659:, and
5092:where
4849:, and
4798:, and
4772:where
4520:(0) =
4290:roots
4284:< 0
4111:, and
4105:> 0
3418:, and
3406:where
3188:, and
2388:where
2234:|
2180:where
2020:kernel
2016:kernel
1610:scalar
1517:where
1072:limits
1044:cosine
903:where
598:People
510:
457:Series
215:Linear
54:Fields
9541:Delay
9487:Order
9214:′ = −
8125:, or
6261:) = 0
3423:<
3001:basis
2461:Let
1550:order
1092:order
736:is a
498:Euler
416:Robin
338:Delay
280:Order
253:Exact
179:Types
47:Scope
9319:ISBN
9301:ISBN
9283:ISBN
9199:′ −
9164:′ −
9082:and
9029:and
9014:and
9002:and
8984:sine
8682:and
8028:Let
7564:for
7508:and
7164:and
6610:the
6337:and
5648:for
5537:and
5017:is
4854:sin(
4843:cos(
4572:and
4554:and
4528:and
4087:and
4062:real
4060:are
4056:and
3882:and
3769:and
2427:′ =
2061:) =
2014:The
1991:) =
1633:ring
1575:Let
1058:and
1040:sine
935:and
732:, a
605:List
9203:= (
9201:hfy
9067:.
8233:exp
8212:exp
8185:of
8123:= 1
7974:")
7573:– 1
6526:As
6379:of
6296:log
5657:– 1
4468:sin
4437:cos
4403:as
4279:If
4191:= 0
4186:If
4100:If
4073:− 4
4052:If
3940:.
3913:sin
3855:cos
3824:by
3634:+ 1
3297:sin
3284:cos
2997:– 1
2449:is
2259:in
2031:= 0
2001:or
1593:or
1572:).
728:In
9874::
9238:^
9228:=
9224:=
9207:)′
9205:hy
9197:hy
9191:hy
9178:=
9168:=
9166:fy
9094:,
9078:,
9043:A
9018:.
8998:,
8994:,
8990:,
8986:,
8982:,
8978:,
8974:,
8970:,
8955:A
8716:.
8705:.
8690:.
8430:.
8268:.
7575:.
6427:=
6165:′(
6148:.
6117:=
5541:)
4884:.
4856:ax
4845:ax
4838:,
4297:βi
4295:±
4199:/2
4096:.
4069:=
3707:ib
3705:–
3697:ib
3695:+
3508:,
3449:−
3445:)(
3181:,
3165:0.
2958:0.
2746:0.
2263:.
2207:,
2055:Ly
2040:,
2033:.
2029:Ly
2011:.
2006:=
2004:Ly
1985:Ly
1596:Lf
1590:Lf
1368:A
1172:A
1146:A
1074:,
1070:,
1054:,
1050:,
1046:,
1042:,
1038:,
1034:,
1022:,
987:.
968:.
9372:e
9365:t
9358:v
9230:e
9226:e
9222:h
9216:f
9212:h
9186:)
9184:x
9182:(
9180:h
9176:h
9170:g
9162:y
8925:1
8919:n
8915:a
8911:,
8905:,
8900:0
8896:a
8873:,
8870:0
8867:=
8864:)
8861:x
8858:(
8855:y
8850:0
8846:a
8842:+
8836:+
8833:)
8830:x
8827:(
8822:)
8819:1
8813:n
8810:(
8806:y
8800:1
8794:n
8790:x
8784:1
8778:n
8774:a
8770:+
8767:)
8764:x
8761:(
8756:)
8753:n
8750:(
8746:y
8740:n
8736:x
8647:.
8644:t
8641:d
8637:)
8634:t
8631:(
8627:b
8623:)
8620:t
8617:(
8612:1
8605:U
8599:x
8592:0
8588:x
8579:)
8576:x
8573:(
8570:U
8567:+
8561:0
8557:y
8552:)
8547:0
8543:x
8539:(
8534:1
8527:U
8523:)
8520:x
8517:(
8514:U
8511:=
8508:)
8505:x
8502:(
8498:y
8477:,
8472:0
8467:y
8462:=
8459:)
8454:0
8450:x
8446:(
8442:y
8411:0
8407:y
8385:,
8382:x
8379:d
8375:)
8372:x
8369:(
8365:b
8361:)
8358:x
8355:(
8350:1
8343:U
8336:)
8333:x
8330:(
8327:U
8324:+
8318:0
8314:y
8309:)
8306:x
8303:(
8300:U
8297:=
8294:)
8291:x
8288:(
8284:y
8273:U
8248:.
8245:)
8242:B
8239:(
8230:A
8227:=
8224:)
8221:B
8218:(
8206:x
8203:d
8199:d
8187:B
8179:U
8162:x
8159:d
8156:A
8150:=
8147:B
8131:A
8127:A
8121:n
8100:)
8097:x
8094:(
8091:U
8075:n
8057:.
8053:u
8049:A
8046:=
8038:u
8011:.
8007:b
8003:+
7999:y
7995:A
7992:=
7984:y
7972:)
7970:x
7968:(
7964:x
7946:j
7943:,
7940:i
7936:a
7909:n
7905:b
7878:,
7873:n
7869:y
7865:)
7862:x
7859:(
7854:n
7851:,
7848:n
7844:a
7840:+
7834:+
7829:1
7825:y
7821:)
7818:x
7815:(
7810:1
7807:,
7804:n
7800:a
7796:+
7793:)
7790:x
7787:(
7782:n
7778:b
7774:=
7767:)
7764:x
7761:(
7753:n
7749:y
7727:n
7723:y
7719:)
7716:x
7713:(
7708:n
7705:,
7702:1
7698:a
7694:+
7688:+
7683:1
7679:y
7675:)
7672:x
7669:(
7664:1
7661:,
7658:1
7654:a
7650:+
7647:)
7644:x
7641:(
7636:1
7632:b
7628:=
7621:)
7618:x
7615:(
7607:1
7603:y
7584:n
7580:n
7571:k
7567:i
7550:,
7545:1
7542:+
7539:i
7535:y
7531:=
7523:i
7519:y
7492:1
7488:y
7484:=
7477:y
7450:k
7446:y
7442:,
7436:,
7431:1
7427:y
7400:)
7397:k
7394:(
7390:y
7386:,
7380:,
7373:y
7369:,
7362:y
7315:.
7310:x
7306:1
7294:+
7289:2
7285:x
7281:=
7278:)
7275:x
7272:(
7269:y
7249:,
7243:=
7240:)
7237:1
7234:(
7231:y
7211:.
7208:x
7204:/
7200:c
7197:+
7192:2
7188:x
7184:=
7181:)
7178:x
7175:(
7172:y
7152:,
7149:c
7146:+
7141:3
7137:x
7133:=
7130:y
7127:x
7108:,
7103:2
7099:x
7095:3
7092:=
7085:)
7081:y
7078:x
7075:(
7055:.
7050:2
7046:x
7042:3
7039:=
7036:y
7033:+
7026:y
7022:x
7000:.
6995:x
6992:c
6987:=
6984:y
6964:,
6959:x
6956:1
6948:=
6943:y
6936:y
6913:0
6910:=
6905:x
6901:)
6898:x
6895:(
6892:y
6886:+
6883:)
6880:x
6877:(
6870:y
6849:.
6846:x
6843:3
6840:=
6835:x
6831:)
6828:x
6825:(
6822:y
6816:+
6813:)
6810:x
6807:(
6800:y
6781:f
6777:F
6773:c
6759:,
6756:x
6753:d
6748:F
6741:e
6737:g
6729:F
6725:e
6721:+
6716:F
6712:e
6708:c
6705:=
6702:y
6682:.
6677:F
6670:e
6666:g
6663:=
6659:)
6653:F
6646:e
6642:y
6638:(
6631:x
6628:d
6624:d
6596:,
6592:)
6587:F
6580:e
6576:(
6568:x
6565:d
6561:d
6555:=
6550:F
6543:e
6539:f
6514:.
6509:F
6502:e
6498:g
6495:=
6490:F
6483:e
6479:f
6476:y
6468:F
6461:e
6453:y
6441:e
6429:e
6425:c
6410:,
6405:F
6401:e
6397:c
6394:=
6391:y
6381:f
6362:x
6359:d
6355:f
6348:=
6345:F
6331:k
6317:,
6314:F
6311:+
6308:k
6305:=
6302:y
6292:,
6289:f
6286:=
6281:y
6274:y
6259:x
6257:(
6255:g
6238:.
6235:)
6232:x
6229:(
6226:g
6223:+
6220:)
6217:x
6214:(
6211:y
6208:)
6205:x
6202:(
6199:f
6196:=
6193:)
6190:x
6187:(
6180:y
6169:)
6167:x
6163:y
6144:n
6140:y
6136:n
6132:u
6128:1
6125:y
6122:1
6119:u
6115:y
6108:n
6104:u
6100:1
6097:u
6083:i
6080:y
6075:f
6069:n
6066:′
6064:u
6060:1
6058:′
6056:u
6051:n
6047:0
6031:.
6026:)
6023:1
6017:n
6014:(
6009:n
6005:y
5996:n
5992:u
5988:+
5982:+
5977:)
5974:1
5968:n
5965:(
5960:1
5956:y
5947:1
5943:u
5939:=
5936:f
5924:n
5920:y
5916:1
5913:y
5908:y
5892:.
5887:)
5884:1
5878:n
5875:(
5870:n
5866:y
5857:n
5853:u
5849:+
5843:+
5838:)
5835:1
5829:n
5826:(
5821:2
5817:y
5808:2
5804:u
5800:+
5795:)
5792:1
5786:n
5783:(
5778:1
5774:y
5765:1
5761:u
5757:+
5752:)
5749:n
5746:(
5741:n
5737:y
5731:n
5727:u
5723:+
5717:+
5712:)
5709:n
5706:(
5701:1
5697:y
5691:1
5687:u
5683:=
5678:)
5675:n
5672:(
5668:y
5655:n
5651:i
5634:)
5631:i
5628:(
5623:n
5619:y
5613:n
5609:u
5605:+
5599:+
5594:)
5591:i
5588:(
5583:1
5579:y
5573:1
5569:u
5565:=
5560:)
5557:i
5554:(
5550:y
5517:,
5512:)
5509:2
5503:n
5500:(
5495:n
5491:y
5482:n
5478:u
5474:+
5468:+
5463:)
5460:2
5454:n
5451:(
5446:2
5442:y
5433:2
5429:u
5425:+
5420:)
5417:2
5411:n
5408:(
5403:1
5399:y
5390:1
5386:u
5382:=
5375:0
5351:n
5347:y
5338:n
5334:u
5330:+
5324:+
5316:2
5312:y
5303:2
5299:u
5295:+
5287:1
5283:y
5274:1
5270:u
5266:=
5259:0
5250:n
5246:y
5237:n
5233:u
5229:+
5223:+
5218:2
5214:y
5205:2
5201:u
5197:+
5192:1
5188:y
5179:1
5175:u
5171:=
5164:0
5150:y
5144:n
5140:u
5136:1
5133:u
5126:n
5122:u
5118:1
5115:u
5110:)
5107:n
5103:y
5099:1
5096:y
5094:(
5080:,
5075:n
5071:y
5065:n
5061:u
5057:+
5051:+
5046:1
5042:y
5036:1
5032:u
5028:=
5025:y
5005:0
5002:=
4999:y
4994:n
4990:a
4986:+
4979:y
4973:1
4967:n
4963:a
4959:+
4953:+
4948:)
4945:1
4939:n
4936:(
4932:y
4926:1
4922:a
4918:+
4913:)
4910:n
4907:(
4903:y
4878:f
4866:a
4862:n
4858:)
4852:x
4847:)
4841:x
4835:e
4832:x
4827:f
4819:f
4815:f
4808:)
4806:x
4804:(
4800:y
4796:x
4792:f
4786:n
4782:a
4778:1
4775:a
4760:,
4757:)
4754:x
4751:(
4748:f
4745:=
4742:)
4739:x
4736:(
4733:y
4728:n
4724:a
4720:+
4717:)
4714:x
4711:(
4704:y
4698:1
4692:n
4688:a
4684:+
4678:+
4675:)
4672:x
4669:(
4664:)
4661:1
4655:n
4652:(
4648:y
4642:1
4638:a
4634:+
4631:)
4628:x
4625:(
4620:)
4617:n
4614:(
4610:y
4599:n
4587:0
4578:2
4575:c
4569:1
4566:c
4560:2
4557:d
4551:1
4548:d
4543:0
4538:2
4535:d
4531:y
4525:1
4522:d
4518:y
4513:)
4511:x
4509:(
4507:y
4489:.
4486:)
4483:)
4480:x
4474:(
4463:2
4459:c
4455:+
4452:)
4449:x
4443:(
4432:1
4428:c
4424:(
4419:x
4412:e
4387:,
4382:x
4379:)
4376:i
4364:(
4360:e
4354:2
4350:c
4346:+
4341:x
4338:)
4335:i
4329:+
4323:(
4319:e
4313:1
4309:c
4293:α
4282:D
4265:.
4260:2
4256:/
4252:x
4249:a
4242:e
4238:)
4235:x
4230:2
4226:c
4222:+
4217:1
4213:c
4209:(
4197:a
4195:−
4189:D
4172:.
4167:x
4160:e
4154:2
4150:c
4146:+
4141:x
4134:e
4128:1
4124:c
4113:β
4109:α
4103:D
4093:2
4090:c
4084:1
4081:c
4075:b
4071:a
4067:D
4058:b
4054:a
4038:.
4035:b
4032:+
4029:r
4026:a
4023:+
4018:2
4014:r
3993:,
3990:0
3987:=
3984:y
3981:b
3978:+
3971:y
3967:a
3964:+
3957:y
3928:)
3925:x
3922:b
3919:(
3908:x
3905:a
3901:e
3895:k
3891:x
3870:)
3867:x
3864:b
3861:(
3850:x
3847:a
3843:e
3837:k
3833:x
3810:x
3807:)
3804:b
3801:i
3795:a
3792:(
3788:e
3782:k
3778:x
3755:x
3752:)
3749:b
3746:i
3743:+
3740:a
3737:(
3733:e
3727:k
3723:x
3703:a
3693:a
3675:.
3654:x
3651:d
3647:d
3632:k
3615:,
3610:x
3603:e
3597:1
3591:k
3587:x
3583:k
3580:=
3576:)
3570:x
3563:e
3557:k
3553:x
3548:(
3543:)
3530:x
3527:d
3523:d
3517:(
3502:P
3498:,
3477:x
3474:d
3470:d
3457:m
3453:)
3451:α
3447:t
3443:t
3441:(
3439:P
3434:m
3430:α
3425:m
3421:k
3416:m
3412:α
3408:k
3394:,
3389:x
3382:e
3376:k
3372:x
3337:.
3332:x
3328:e
3324:x
3320:,
3315:x
3311:e
3306:,
3303:x
3293:,
3290:x
3264:.
3259:x
3255:e
3251:x
3247:,
3242:x
3238:e
3233:,
3228:x
3225:i
3218:e
3213:,
3208:x
3205:i
3201:e
3190:1
3185:i
3183:−
3179:i
3162:=
3159:1
3156:+
3153:z
3150:2
3142:2
3138:z
3134:2
3131:+
3126:3
3122:z
3118:2
3110:4
3106:z
3085:0
3082:=
3079:y
3076:+
3069:y
3065:2
3055:y
3051:2
3048:+
3041:y
3037:2
3027:y
2995:n
2991:x
2978:n
2955:=
2950:n
2946:t
2940:n
2936:a
2932:+
2926:+
2921:2
2917:t
2911:2
2907:a
2903:+
2900:t
2895:1
2891:a
2887:+
2882:0
2878:a
2852:n
2848:t
2842:n
2838:a
2834:+
2828:+
2823:2
2819:t
2813:2
2809:a
2805:+
2802:t
2797:1
2793:a
2789:+
2784:0
2780:a
2766:α
2761:e
2743:=
2738:x
2731:e
2725:n
2715:n
2711:a
2707:+
2701:+
2696:x
2689:e
2683:2
2673:2
2669:a
2665:+
2660:x
2653:e
2644:1
2640:a
2636:+
2631:x
2624:e
2618:0
2614:a
2603:α
2598:e
2588:n
2584:a
2580:0
2577:a
2562:0
2559:=
2554:)
2551:n
2548:(
2544:y
2538:n
2534:a
2530:+
2524:+
2517:y
2511:2
2507:a
2503:+
2496:y
2490:1
2486:a
2482:+
2479:y
2474:0
2470:a
2455:e
2452:c
2446:e
2441:n
2435:f
2429:f
2425:f
2419:e
2402:n
2398:a
2394:1
2391:a
2376:0
2373:=
2368:)
2365:n
2362:(
2358:y
2352:n
2348:a
2344:+
2338:+
2331:y
2325:2
2321:a
2317:+
2310:y
2304:1
2300:a
2296:+
2293:y
2288:0
2284:a
2261:I
2257:x
2252:k
2248:)
2246:x
2244:(
2241:n
2237:a
2230:k
2226:I
2220:n
2216:a
2212:0
2209:a
2205:b
2200:I
2194:n
2190:c
2186:1
2183:c
2168:,
2165:)
2162:x
2159:(
2154:n
2150:S
2144:n
2140:c
2136:+
2130:+
2127:)
2124:x
2121:(
2116:1
2112:S
2106:1
2102:c
2098:+
2095:)
2092:x
2089:(
2084:0
2080:S
2069:)
2067:x
2065:(
2063:b
2059:x
2057:(
2050:n
2046:L
2038:n
2008:b
1999:)
1997:x
1995:(
1993:b
1989:x
1987:(
1980:y
1964:.
1961:)
1958:x
1955:(
1952:b
1949:=
1946:y
1943:L
1923:)
1920:x
1917:(
1914:b
1911:=
1906:)
1903:n
1900:(
1896:y
1892:)
1889:x
1886:(
1881:n
1877:a
1873:+
1867:+
1860:y
1856:)
1853:x
1850:(
1845:2
1841:a
1837:+
1830:y
1826:)
1823:x
1820:(
1815:1
1811:a
1807:+
1804:y
1801:)
1798:x
1795:(
1790:0
1786:a
1765:,
1757:n
1753:x
1749:d
1743:n
1739:d
1733:)
1730:x
1727:(
1722:n
1718:a
1714:+
1708:+
1702:x
1699:d
1695:d
1690:)
1687:x
1684:(
1679:1
1675:a
1671:+
1668:)
1665:x
1662:(
1657:0
1653:a
1649:=
1646:L
1602:)
1600:X
1598:(
1585:f
1581:L
1577:L
1566:)
1564:x
1562:(
1559:n
1555:a
1546:n
1542:)
1540:x
1538:(
1535:n
1531:a
1527:x
1525:(
1523:0
1520:a
1505:,
1497:n
1493:x
1489:d
1483:n
1479:d
1473:)
1470:x
1467:(
1462:n
1458:a
1454:+
1448:+
1442:x
1439:d
1435:d
1430:)
1427:x
1424:(
1419:1
1415:a
1411:+
1408:)
1405:x
1402:(
1397:0
1393:a
1363:n
1342:n
1338:i
1332:n
1328:x
1314:1
1310:i
1304:1
1300:x
1288:n
1284:i
1280:+
1274:+
1269:1
1265:i
1228:i
1224:x
1220:d
1214:i
1210:d
1197:i
1187:i
1178:i
1103:)
1101:x
1099:(
1097:b
966:x
962:y
957:y
953:y
944:)
942:x
940:(
938:b
932:)
930:x
928:(
925:n
921:a
916:)
914:x
912:(
910:0
907:a
891:)
888:x
885:(
882:b
879:=
874:)
871:n
868:(
864:y
860:)
857:x
854:(
849:n
845:a
841:+
831:y
827:)
824:x
821:(
816:2
812:a
808:+
801:y
797:)
794:x
791:(
786:1
782:a
778:+
775:y
772:)
769:x
766:(
761:0
757:a
717:e
710:t
703:v
516:)
512:(
34:.
20:)
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