Given the experimental results for the ideal gas, I want to recover the internal energy expressed in its natural variables as:
$$U(S,V,N) = \alpha e^{\frac{S}{N c_V}} V^{\frac{c_V-c_p}{c_V}} N^{\frac{c_p}{c_V}}$$
with $\alpha$ as some constant and $c_V,c_p$ the (intensive) heat capacities. My starting point are the two equations:
$$pV=(c_P-c_V)NT,$$ $$U=c_V NT.$$
These completely characterize the system, and so does any thermodynamic potential expressed in its 'natural' variables, so one should follow from the other. To recover the experimental facts from $U(S,V,N)$ is simple enough, but I'm interested in the other direction. Unfortunately I was unable to find this derivation anywhere - I found the formula mentioned on wikipedia but I can't get my hands on the books cited.
So, how to do it? I'm not at all interested in any statistical physics considerations, only pure thermodynamics. Preferably without any elaborate tricks, I believe this must be doable in a methodical manner with just straightforward (if tedious?) math, i.e. using with all the multivariable calculus identities (Jacobians etc.), and this is the method I'd like to see.