Newton's Law in the two coordinate directions says
That system looks horrible to solve, but the thing is that I can't come up with a good generalized coordinate for this problem. The best I can offer is to put an angular variable at the focus of the parabola.
However, things aren't all bleak. All we need is the coefficients of the expansion:
Presumably if h is small then A = 0, the coefficients of the powers of x higher than 1 are small, and B is negative. In that case
which is the harmonic oscillator equation. I leave it to you to check that these conditions come out. (I haven't checked.)
Edit: After seeing the original posting of this problem it is clear that the problem is requiring an exact, not approximate solution. The only way I can think of to get this is to use the Lagrangian technique. But again, I can't think of a good generalized coordinate here.