# Second order PDE with coupled nonlinear coefficients

I am trying to separate and solve this PDE:
$$\frac{\partial^2{S(u,v)}}{\partial{u}^2}+\frac{\partial^2{S(u,v)}}{\partial{v}^2}-\left[\frac{a^2}{2 \cosh(u) \cos(v)}\right] S(u,v) =0$$
where $(u,v)$ are orthogonal curvilinear coordinates associated with a bipolar conformal mapping from Cartesian coordinates. Can someone suggest an analytic method to solve this in a separated closed-form that will allow me to satisfy boundary conditions on contours of constant $(u)$?

I have attempted a separation of variables approximation with $(v)$ held constant about some constant contour $(v=v_0)$:
$$\frac{\partial^2{T(u)}}{\partial{u}^2}-\left[\frac{\gamma^2}{ \cosh(u)} +\beta^2 \right] T(u) =0$$
where: $$\gamma^2=\frac{a^2}{2 \cos(v_0)}$$
but am unable to find a solution for this ODE.