Thermal Phys: Show (du/dP)|t = Pkv - (cp - cv)K/B

[StarTiger]

For some reason this proof in thermal is tripping me up.

Homework Statement

Show that (du/dP)|t = Pkv - (cp - cv)K/B (problem 4-4 in Carter or 1.14 in Cheng)

Homework Equations

B = 1/v(R/P) = 1/T
k= -1/v(-RT/p^2) = 1/P

The Attempt at a Solution

Start with 1st law dq=(du + Pdv) and substitute...eventually:
Get to du/dP|T dP - (du/dt)|p dt - PBvdT +PkvdP
Divide by dP

Now is where I'm stuck...

 

[Mapes]

You want to prove this generally, not just for the case of the ideal gas. Do you know the definitions of $C_P$ and $C_V$?