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DEtools

  

FindODE

  

find a linear ODE for a holonomic function

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

FindODE(f, v, maxorder)

Parameters

f

-

expression

v

-

either the dependent variable, of the form yx, or a list of two names Dx,x

maxorder

-

(optional) posint; maximal order of the ODE. Default value: 6

Description

• 

The input f should be a holonomic function, in other words, a function that satisfies a linear ODE with rational function coefficients. The FindODE function tries to find such an ODE.

• 

If FindODE fails to find a linear ODE, then FAIL is returned.

• 

When using the FindODEf,yx calling sequence, the ODE will be returned in terms of the dependent variable yx, where x is the independent variable.

• 

When using the FindODEf,Dx,x calling sequence, the result will be given in differential operator form, that is, as a polynomial in the differential operator Dx whose coefficients are polynomials in the independent variable x.

• 

The second argument v can be omitted if the environment variable _Envdiffopdomain is assigned a list of two names, in which case the result will be given in differential operator notation.

• 

The resulting ODE will be cleared of denominators, that is, its coefficients are polynomials in the independent variable x without common factors.

• 

By default, FindODE incrementally searches for an ODE up to order 6. This maximal order can be overridden by specifying the optional third argument, maxorder.

Examples

withDEtools:

FindODEcosx,yx

yx+2ⅆⅆxyx+4xⅆ2ⅆx2yx

(1)

FindODEⅇx+x,yx

2x+1yx+4x21ⅆⅆxyx+4x22xⅆ2ⅆx2yx

(2)

The hypergeometric 2F1 equation.

FindODEhypergeoma,b,c,x,yx

abyx+xa+xbc+xⅆⅆxyx+x2xⅆ2ⅆx2yx

(3)

The tangent function is not holonomic, so the result is FAIL.

FindODEtanx,yx

FAIL

(4)

Operator notation.

FindODEcosx,Dx,x

4xDx2+2Dx+1

(5)

The following example is a generating function from the Online Encyclopedia of Integer Sequences (http://oeis.org/A151357).

ogfx2∫∫2hypergeom34,54,2,64x31+x14x2214x232ⅆxⅆx

ogf2hypergeom34,54,2,64x31+x4x2+124x2+132ⅆxⅆxx2

(6)

LFindODEogf,Dx,x

L192x10+640x9+880x8+656x7+244x6+16x57x43x3Dx4+3072x9+9792x8+13344x7+10016x6+3632x5+244x486x336x2Dx3+13824x8+42048x7+56832x6+42816x5+15072x4+1068x3264x2108xDx2+18432x7+53376x6+71616x5+53952x4+18336x3+1416x2180x72Dx+4608x6+12672x5+16896x4+12672x3+4128x2+360x

(7)

DFactorL,Dx,x

192x10+640x9+880x8+656x7+244x6+16x57x43x3Dx2+21152x7+3616x6+4912x5+3696x4+1328x3+90x229x12Dxx192x7+640x6+880x5+656x4+244x3+16x27x3+22880x7+8480x6+11408x5+8592x4+2956x3+222x237x15x2192x7+640x6+880x5+656x4+244x3+16x27x3,Dx+2x,Dx+2x

(8)

mapdegree,,Dx

2,1,1

(9)

The first order factors of L come from the integrals. The second order factor comes from the hypergeometric 2F1 function.

Summing special functions preserves the order if they are contiguous (parameter differences are integers).

FindODEBesselI0,x+xBesselI2,x,yx

x42x34x23xyx+x3+x+4ⅆⅆxyx+x4+2x3+x2+2xⅆ2ⅆx2yx

(10)

Products of two contiguous second-order special functions satisfy third order equations.

FindODEBesselI0,xBesselI3,x,Dx,x

24x4+64x3Dx3+84x3+288x2Dx2+24x3130x2+16xDx24x2179x216

(11)

FindODEhypergeom13,23,1,x2hypergeom13,13,1,xhypergeom43,23,1,x,Dx,x

9x527x3+18x2Dx3+45x4+54x3135x2+36xDx2+38x3+110x294xDx+2x2+18x2

(12)

Non-contiguous examples:

FindODEKummerM14,1,x+KummerM14,1,x,Dx,x

16x2Dx4+32x2+64xDx3+16x280x+32Dx2+16x16Dx1

(13)

FindODELegendreP14,xLegendreP12,x,yx

495yx+1440xⅆⅆxyx+5600x21504ⅆ2ⅆx2yx+2560x32560xⅆ3ⅆx3yx+256x4512x2+256ⅆ4ⅆx4yx

(14)

dsolve

yx=_C1LegendreP12,xLegendreQ14,x+_C2LegendreP14,xLegendreP12,x+_C3LegendreQ12,xLegendreP14,x+_C4LegendreQ12,xLegendreQ14,x

(15)

Compatibility

• 

The DEtools[FindODE] command was introduced in Maple 2019.

• 

For more information on Maple 2019 changes, see Updates in Maple 2019.

See Also

DEtools

gfun[holexprtodiffeq]

LinearOperators

PDEtools[dpolyform]