The general form of the ellipsoidal ODE is given by the following:

with(DEtools,odeadvisor);

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

ellipsoidal_ode := diff(y(x),x,x) = (a+b*k^2*sin(x)^2+q*k^4*sin(x)^4)*y(x);

$\mathrm{ellipsoidal\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\left(a+b{k}^2{\sin \left(x\right)}^2+q{k}^4{\sin \left(x\right)}^4\right)y\left(x\right)$

odeadvisor(ellipsoidal_ode);

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

See Arscott, "The Land Beyond Bessel: A Survey of Higher Special Functions".