dperiodic_sols - Maple Help

DEtools

 dperiodic_sols
 find doubly-periodic solutions of a linear ODE

 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 $\mathrm{sn}$, $\mathrm{cn}$, and $\mathrm{dn}$ 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

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{alias}\left(P=\mathrm{WeierstrassP}\left(x,\mathrm{g2},\mathrm{g3}\right),\mathrm{Pp}=\mathrm{WeierstrassPPrime}\left(x,\mathrm{g2},\mathrm{g3}\right),\mathrm{sn}=\mathrm{JacobiSN}\left(x,k\right),\mathrm{cn}=\mathrm{JacobiCN}\left(x,k\right),\mathrm{dn}=\mathrm{JacobiDN}\left(x,k\right)\right):$

Special case of Lame equation (n=1).

 > $\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)-\left(2P+B\right)y\left(x\right):$
 > $\mathrm{dperiodic_sols}\left(\mathrm{ode},y\left(x\right)\right)$
 $\left[{{ⅇ}}^{{-}\frac{\sqrt{{4}{}{{B}}^{{3}}{-}{B}{}{\mathrm{g2}}{-}{\mathrm{g3}}}{}\left({\int }\frac{{1}}{{B}{-}{P}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{{2}}}{}\sqrt{{B}{-}{P}}{,}{{ⅇ}}^{\frac{\sqrt{{4}{}{{B}}^{{3}}{-}{B}{}{\mathrm{g2}}{-}{\mathrm{g3}}}{}\left({\int }\frac{{1}}{{B}{-}{P}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{{2}}}{}\sqrt{{B}{-}{P}}\right]$ (1)

Kamke 2.74

 > $k≔3:$
 > $\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+\frac{{k}^{2}\mathrm{sn}\mathrm{cn}}{\mathrm{dn}}\mathrm{diff}\left(y\left(x\right),x\right)+9{\mathrm{dn}}^{2}y\left(x\right):$
 > $\mathrm{dperiodic_sols}\left(\mathrm{ode},y\left(x\right)\right)$
 $\left[{-}\frac{{4}{}{{\mathrm{JacobiSN}}{}\left({x}{,}{3}\right)}^{{3}}}{{3}}{+}{\mathrm{JacobiSN}}{}\left({x}{,}{3}\right){,}\left({-}\frac{{4}{}{{\mathrm{JacobiSN}}{}\left({x}{,}{3}\right)}^{{3}}}{{3}}{+}{\mathrm{JacobiSN}}{}\left({x}{,}{3}\right)\right){}\left(\frac{{18}{}{\mathrm{JacobiSN}}{}\left({x}{,}{3}\right){}{\mathrm{JacobiCN}}{}\left({x}{,}{3}\right)}{{{\mathrm{JacobiDN}}{}\left({x}{,}{3}\right)}^{{2}}{+}\frac{{23}}{{4}}}{+}\frac{{9}{}{\mathrm{JacobiSN}}{}\left({x}{,}{3}\right){}{\mathrm{JacobiCN}}{}\left({x}{,}{3}\right)}{{{\mathrm{JacobiDN}}{}\left({x}{,}{3}\right)}^{{2}}{-}{1}}\right)\right]$ (2)

Kamke 2.73 (Only doubly periodic solutions).

 > $\mathrm{ode}≔\left(\mathrm{Pp}+{P}^{2}\right)\mathrm{diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+\left({P}^{3}-P\mathrm{Pp}-\mathrm{diff}\left(P,\mathrm{}\left(x,2\right)\right)\right)\mathrm{diff}\left(y\left(x\right),x\right)+\left({\mathrm{Pp}}^{2}-{P}^{2}\mathrm{Pp}-P\mathrm{diff}\left(P,\mathrm{}\left(x,2\right)\right)\right)y\left(x\right):$
 > $\mathrm{_EnvDperiodicOnly}≔\mathrm{true}$
 ${\mathrm{_EnvDperiodicOnly}}{≔}{\mathrm{true}}$ (3)
 > $\mathrm{dperiodic_sols}\left(\mathrm{ode},y\left(x\right)\right)$
 $\left[{P}\right]$ (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.