Well, this does make quite a difference. The latest equation is homogeneous. A substitution

p = vq(v)

leads to a separable equation for the function q. Which, in principle, can be solved by common integration. I have not checked that the resulting integral is simple or expressible in terms of elementary functions.

Also, at an academic institution, there must be available some kind of symbolic integration package: Mathematica, Matlab, or something more specialized. I would check.