I have recently got stuck on this ODE in my studies of rogue waves on elliptic backgrounds:
$$ y'' + q(x) y' + w(x) y = 0 $$ where $$ q(x) = m \frac{sn(x|m)\,cn(x|m)}{dn(x|m)}, \\ w(x) = \left(\lambda^2 + dn(x|m)^2 - i m \lambda \frac{sn(x|m)\,cn(x|m)}{dn(x|m)} \right) $$ The parameter $\lambda \in \mathbb{C}$ and $m \in [0,1]$ is the elliptic parameter of the Jacobi elliptic functions $\text{dn}(.|.), \, \text{cn}(.|.)$ and $\text{sn}(.|.)$. I am only interested in $x \in \mathbb{R}$.
Note that (according to Mathematica), the period of $q(x)$ is $4K(m)$ where $K$ is the complete elliptic integral of the first kind. The real and imaginary parts of $w(x)$ have the same period as $q(x)$.
The initial value problem: $$ y(0) = 2i\sin(\alpha),\\ y'(0) = 2(\lambda i \sin(\alpha) + i \cos(\beta)) $$ where $\alpha, \beta \in \mathbb{C}$ are some parameters that depend on $\lambda$ in some complicated way.
Are there any specific tricks to help me find an analytical solution for this ODE? Does anyone have any pointers? I have checked a few handbooks on Jacobi elliptic functions but haven't had much luck, I have no formal training in elliptic functions and not too much in ODEs.
I have solved it numerically but I'm wondering if it's possible to solve it analytically for completeness.
EDIT: I have taken a look at this paper but it's a little too complicated for me. However, based on the transformation they present in equation 15:
$$ y(x) = \exp\left(-\frac{1}{2}\int q(x) dx\right) $$ we can reduce the first equation to: $$ u''(x) - r(x) u(x) = 0 $$ where $$ r(x) = \frac{1}{2} q'(x) + \frac{1}{4} q^2(x) - w(x) $$ According to Mathematica: $$ r(x) = \frac{3 (m-1)+\text{dn}(x|m) \left(-\text{dn}(x|m) \left(3 \text{dn}(x|m)^2+4 \lambda ^2+m-2\right)+4 i m \text{cn}(x|m) \text{sn}(x|m)\right)}{4 \text{dn}(x|m)^2} $$
I have expanded this in a Laurent series as suggested in the comments, but I am not sure how to properly manipulate the product of the series of $r(x) u(x)$ on the RHS of $u'' = r(x) u$.