I am interested in solving a differential equation of the form
$$\left(\frac{dx}{dr}\right)^{\!2}=F(x,y)+G(x)\left(\frac{dy}{dr}\right)^{\!2}$$
The reason most of my tricks don't work are that $F$ and $G$ are relatively complicated, and I'm not sure how to deal with the two nonlinear derivatives.
I do know that there are numerical methods that might work, for instance, I could take a derivative, set $a=dx/dr$, $b=dy/dr$ and form a linear system, but I'd like to check analytic approaches first.
EDIT: Following recommendations from comments, I'll post the precise forms of $F$ and $G$ - hope it doesn't make people ignore my question!
\begin{align*} F(x,y)&=\frac{1-b^2}{|\wp(z,\omega_1,\omega_2)|^2}(1+\cos(x))^2 \\ G(x)&=-\sin^2(x), \end{align*}
where $b$ is a constant and $\wp(z,\omega_1,\omega_2)$ is the Weierstrass function with periods $\omega_1$ and $\omega_2$, with complex domain being mapped to the real domain via $$z=\tan(x/2)\exp(iy).$$