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.

(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

I followed the same process as you did, up until the point where you did partial fractions, while I completed the square on the denominator of the integrand. (Also, I forgot a factor of 9 in my integrand, which of course would not affect its integrability.) However, after that, the forms of the integral look quite different, although they must be equivalent. Yours appears to be nicer.

--Stuart Anderson