Where does the 4 come from in the derivative of an integral?

[juice34]

Homework Statement

Can someone help me take the derivative of the integral
u(1/r)((d/dr)[(r)(dV/dr)])=P

Homework Equations

The Attempt at a Solution

my attempt yields V=(Pr^2)/(2u)+C(1), which is not right. The actual answer is V=(Pr^2)/4u+C(1)ln(r)+C(2). I am having trouble finding out where the 4 comes from could someone please explain to me what is going on. Thank YOU

 

[LCKurtz]

Is u a function of r? If not why write u(1/r) instead of just (u/r)? Are u and P constants? I think I know the answers to those questions but you should pose your question more carefully. Rewriting your equation using ' for d/dr:

(rV')' = Pr/u

Integrate:

rV' = Pr²/(2u) + C

Divide both sides by r and integrate again.

 

[juice34]

LCKurtz u and P are constant. And I am not sure i follow this part, (rV')' = r/u. Is it equal to d/dx(r(dV/dr))

 

[LCKurtz]

I accidently left off the P, which I just edited to correct.

(rV')' is d/dr (r dV/dr)

No x in there. Its much neater to write with primes.

 

[juice34]

LCKurtz i cannot thank you enough, you are a excellent help!