Determination of the equation of isentropic processes

[young_qubit]

I have been given a fundamental equation of a system as
$$u = \frac{s^4}{v^2}$$
After writing down the 3 equations of state, namely:
$$T = 4\frac{S^3}{VN}$$
$$P = \frac{1}{2}\frac{S^4}{V^{3}N}$$
$$\mu = -\frac{S^4}{VN^{2}}$$

I need to determine the equation of isentropic (dS = 0) processes on the P-V diagram. I understand that the relationship should only contain P, v (plus whatever constants), but I'm not sure what to do now. I was thinking that I should put these values into
$$dQ = dU + PdV$$
where I know $dQ = TdS = 0$ by above definition, and assuming mols constant. Which would give me
$$dU = -PdV \,\rightarrow\,\frac{1}{2}\frac{S^4}{V^{3}N}$$

but I'm not confident that's right. Looking for some suggestions, thanks.

 

[njoysci]

refer to the analysis further down the article

http://en.wikipedia.org/wiki/Internal_energy