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dperiodic_sols

  

find doubly-periodic solutions of a linear ODE

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

dperiodic_sols(lode, v)

dperiodic_sols(coeff_list, x)

Parameters

lode

-

linear ODE in diff form

v

-

dependent variable of lode

coeff_list

-

list of coefficients of a linear ODE; specified in order of increasing differential order

x

-

independent variable of a linear ODE

Description

• 

The dperiodic_sols function seeks closed form solutions of linear ODEs having doubly-periodic coefficients. It returns either one or more doubly-periodic solutions to the linear ODE or provides a proof that no such solution exists.

  

In the case of an order two linear ODE, the dperiodic_sols function also seeks a general solution in terms of solutions that are doubly-periodic or doubly-periodic of the second kind.

• 

The dperiodic_sols function returns, if possible, a list of one or more independent solutions. To find only doubly periodic solutions, set the environment variable _EnvDperiodicOnly to true.

• 

The dperiodic_sols function recognizes doubly-periodic functions that are rational in the Weierstrass P and P' functions, or rational in the Jacobi sn, cn, and dn functions.

• 

The dperiodic_sols(lode, v) and dperiodic_sols(coeff_list, x) calling sequences are equivalent.

  

The dperiodic_sols(coeff_list, x) calling sequence is convenient for programming.

Examples

withDEtools:

aliasP=WeierstrassPx,g2,g3,Pp=WeierstrassPPrimex,g2,g3,sn=JacobiSNx,k,cn=JacobiCNx,k,dn=JacobiDNx,k:

Special case of Lame equation (n=1).

odeⅆ2ⅆx2yx2P+Byx:

dperiodic_solsode,yx

ⅇ4B3Bg2g31BPⅆx2BP,ⅇ4B3Bg2g31BPⅆx2BP

(1)

Kamke 2.74

k3:

odeⅆ2ⅆx2yx+k2sncnⅆⅆxyxdn+9dn2yx:

dperiodic_solsode,yx

4JacobiSNx,333+JacobiSNx,3,4JacobiSNx,333+JacobiSNx,318JacobiSNx,3JacobiCNx,3JacobiDNx,32+234+9JacobiSNx,3JacobiCNx,3JacobiDNx,321

(2)

Kamke 2.73 (Only doubly periodic solutions).

odePp+P2ⅆ2ⅆx2yx+P3PPp2x2Pⅆⅆxyx+Pp2P2PpP2x2Pyx:

_EnvDperiodicOnlytrue

_EnvDperiodicOnlytrue

(3)

dperiodic_solsode,yx

P

(4)

References

  

Burger, R.; Labahn, G.; and van Hoeij, M. "Closed form solutions of linear odes having elliptic functions as coefficients." Proceedings of ISSAC'04, Santander, Spain, ACM Press, (2004): 58-64.

See Also

alias

dcoeffs

DEtools

dsolve

Environment Variables

Jacobi functions

Weierstrass functions