If $U=f(P,V,T)$ be the internal energy of a gas that obeys the ideal gas law $PV=nRT$, ($n$ and $R$ constants) and$(\frac{∂U}{∂T})_v+(\frac{∂U}{∂P})_v=α(\frac{∂U}{∂T})+β(\frac{∂U}{∂P})$then α+β equals
(A)$2+\frac{v}{nR}+\frac{nR}{v}$
(B)$1+\frac{v}{nR}$
(C)$1+\frac{n}{Rv}$
(D) 1
Let us take $U\equiv PV-nRT$,then $(\frac{∂U}{∂T})_v+(\frac{∂U}{∂P})_v=α(\frac{∂U}{∂T})+β(\frac{∂U}{∂P})\implies -nR+V=-\alpha nR +\beta V$
on comparing coefficient of both sides we get $\alpha=1$ and $\beta =1$.So,$\alpha +\beta =2$
But it does not matches with none of the option.
Where i'm commiting mistake??