The general forms of the elliptic ODEs are given by the following:

elliptic_I_ode := diff(x*(1-x^2)*diff(y(x),x),x)-x*y(x)=0;

$\mathrm{elliptic\_I\_ode}≔\left(-x^2+1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-2x^2\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+x\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-xy\left(x\right)=0$

elliptic_II_ode := (1-x^2)*diff(x*diff(y(x),x),x)+x*y(x)=0;

$\mathrm{elliptic\_II\_ode}≔\left(-x^2+1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)+x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\right)+xy\left(x\right)=0$

See Gradshteyn and Ryzhik, "Tables of Integrals, Series and Products", p. 907. The solution to this type of ODE can be expressed in terms of the EllipticK and EllipticCK functions.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{dsolve}\left(\mathrm{elliptic\_I\_ode}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{EllipticK}\left(x\right)+\mathrm{c\_\_2}\mathrm{EllipticCK}\left(x\right)$

$\mathrm{dsolve}\left(\mathrm{elliptic\_II\_ode}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{EllipticE}\left(x\right)+\mathrm{c\_\_2}\left(\mathrm{EllipticCE}\left(x\right)-\mathrm{EllipticCK}\left(x\right)\right)$

$\mathrm{odeadvisor}\left(\mathrm{elliptic\_I\_ode}\right)$

$\left[\left[\mathrm{\_elliptic}\,\mathrm{\_class\_I}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{elliptic\_II\_ode}\right)$

$\left[\left[\mathrm{\_elliptic}\,\mathrm{\_class\_II}\right]\right]$