I looked at this for a while yesterday, and made little progress, eventually concluding that this differential equation was not amenable to any of the tricks in my repertoire. The revised version is, as Alex says, homogeneous. I looked at the form you get if you make the substitution p = vq, and the resulting integrands do simplify in a nice way if you make the further substitution r = q-1/3, which comes from completing the square on the denominator after variable separation. The result is

(1-6r)dr/(r^2-1/9) = dv/v.

The left side integrates easily to one trig term and one reciprocal term. This will give an implicit function for p in terms of v, which would be rather hard (I think) to solve explicitly for p. On the other hand, you can perhaps get v explicitly as a function of p, although the substitution p=qv will make this awkward. In any case, the differential equation is "solved" in the formal sense that the expression relating p and v now contains no derivatives, although extracting the value of p for a given v will still require solving a rather messy algebraic equation. I suspect that plotting the result numerically will be the best approach.

Hope that helps!

Stuart Anderson