Determine equation of state from entropy

[A_B]

Homework Statement

The entropy is given as
$$S=a \left(VNU \right)^{\frac{1}{3}}$$
Find the equation of state.

2. The attempt at a solution
I've tried writing $dS$ in terms of $dP$ and $dT$ then using the fact that $dS$ is a perfect differential equate the partial derivatives of the terms. This got me nowhere. I also tried using Maxwell relations.
$$\left( \frac{\partial S}{\partial V} \right) _T = \left( \frac{\partial P}{\partial T} \right) _V$$

Thanks
Alex

 

[A_B]

How about I use the definition of temperature?
$$\frac{1}{T} = \left( \frac{\partial S}{\partial U} \right) = \frac{a}{3} \left( \frac{VT}{U^2} \right)^{1/3}$$