[UPSC CSE 2022] Thermodynamics (Q28, 15M)

A gas of interacting atoms has an equation of state and heat capacity at constant volume given by the expressions p(T,V)=aT^{1/2}+bT^3+cV^{-2} C_v(T,V)=dT^{1/2}+eT^2V+fT^{1/2} where a through f are constants which are independent of T and V. Find the differential of the internal energy dU(T,V) in terms of dT and dV.