Simple Partial Differential Equation

1. Oct 12, 2014

jaydnul

1. The problem statement, all variables and given/known data
This is actually an electromagnetism problem but all the physics is done, I just don't remember how to solve the PDE:
$\frac{d^2V}{dr^2}=-\frac{2}{r}\frac{dV}{dr}$
The d's should be del's, just don't know how to do that...

2. Relevant equations

Not sure.

3. The attempt at a solution

Don't know where to start.

2. Oct 12, 2014

Dick

There are no other independent variables in the problem besides r. Assume V can be written as a product of functions of all the independent variables. I.e. it's separable. V(r)=u(r)*v(theta)*w(phi). The treat it as an ODE. Do you remember those? I would try putting dV/dr=u(r) and separating variables.

Last edited: Oct 12, 2014
3. Oct 12, 2014

jaydnul

Ahh I think that's just what I needed. So

$V_{rr}=-k^2u(r)$ where k=sqrt{2/r}
$u(r)=Ae^{ikr}+Be^{-ikr}$
$V(r)=\frac{-iA}{k}e^{ikr}+\frac{iB}{k}e^{-ikr}$
then sub in for k

Is that correct?

4. Oct 12, 2014

Dick

Not really. If u(r)=dV/dr then the equation becomes du/dr=(-2/r)u. Doesn't it? Solve that.

5. Oct 12, 2014

jaydnul

Ahh, I'm an idiot. $u=r^{-2}$

Thanks Dick!

6. Oct 12, 2014

Dick

You're welcome but don't forget the arbitrary constant when you integrate.

7. Oct 12, 2014

Ray Vickson

Note that
$$\frac{d}{dr}\left( r^k \frac{dF(r)}{dr} \right) = r^k \frac{d^2F(r)}{dr^2}+ k r^{k-1} \frac{dF(r)}{dr}$$
Can you see how that fact helps?