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.

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

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

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-\left(2P+B\right)y\left(x\right)\:$

$\mathrm{dperiodic\_sols}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left[{\ⅇ}^{-\frac{\sqrt{4B^3-B\mathrm{g2}-\mathrm{g3}}\left(\int \frac{1}{B-P}\ⅆx\right)}{2}}\sqrt{B-P}\,{\ⅇ}^{\frac{\sqrt{4B^3-B\mathrm{g2}-\mathrm{g3}}\left(\int \frac{1}{B-P}\ⅆx\right)}{2}}\sqrt{B-P}\right]$

$k≔3\:$

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+\frac{k^2\mathrm{sn}\mathrm{cn}}{\mathrm{dn}}\mathrm{diff}\left(y\left(x\right)\,x\right)+9{\mathrm{dn}}^{2}y\left(x\right)\:$

$\mathrm{dperiodic\_sols}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left[-\frac{4{\mathrm{JacobiSN}\left(x\,3\right)}^3}{3}+\mathrm{JacobiSN}\left(x\,3\right)\,\left(-\frac{4{\mathrm{JacobiSN}\left(x\,3\right)}^3}{3}+\mathrm{JacobiSN}\left(x\,3\right)\right)\left(\frac{18\mathrm{JacobiSN}\left(x\,3\right)\mathrm{JacobiCN}\left(x\,3\right)}{{\mathrm{JacobiDN}\left(x\,3\right)}^2+\frac{23}{4}}+\frac{9\mathrm{JacobiSN}\left(x\,3\right)\mathrm{JacobiCN}\left(x\,3\right)}{{\mathrm{JacobiDN}\left(x\,3\right)}^2-1}\right)\right]$

$\mathrm{ode}≔\left(\mathrm{Pp}+P^2\right)\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+\left(P^3-P\mathrm{Pp}-\mathrm{diff}\left(P\,\mathrm{`\$`}\left(x\,2\right)\right)\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+\left({\mathrm{Pp}}^2-P^2\mathrm{Pp}-P\mathrm{diff}\left(P\,\mathrm{`\$`}\left(x\,2\right)\right)\right)y\left(x\right)\:$

$\mathrm{\_EnvDperiodicOnly}≔\mathrm{true}$

$\mathrm{\_EnvDperiodicOnly}≔\mathrm{true}$

$\mathrm{dperiodic\_sols}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left[P\right]$

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.