Differential equation problem (originally kinematics)

A point mass ‘A’ is kept at the origin. Another B is kept at the x axis at x = H. Another C is kept at distance H from origin and distance h from B. A, B and C thus form an isosceles triangle with vertex A. Given that A's mass >> B's mass >> C's mass; Newton’s gravitational law governs the bodies. C will have motion because of both A and B while B will have motion because of A only.

Find:

The equation of path of C in any coordinate system given that B and C have zero initial velocities.

This problem after a bit of physics will reduce to a problem of differential equations. I applied r, $\vartheta$ coordinate system and got two differential equations in r, $\vartheta$ and both their single and double time derivatives. A very complex eqn indeed!

Solve or suggest a way to solve this problem, please!