You can integrate once immediately by reducing the order. You have no x(t)'s or y(t)'s in there. Once you do that, you might be able to use a polar coordinate transformation to simplify.
I can see what you did there. Multiplying both equations by what you need in order to get the same thing on the RHS's. That's fine. Now what if you were to divide the entire equation by uv?
[Edit]: I see you anticipated me. Excellent. What can you say now?