Simple Partial Differential Equation

[jaydnul]

Homework Statement

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...

Homework Equations

Not sure.

The Attempt at a Solution

[/B]
Don't know where to start.

 

[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.

 

[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?

 

[Dick]

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?

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

 

[jaydnul]

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

Thanks Dick!

 

[Dick]

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

Thanks Dick!

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

 

[Ray Vickson]

Homework Statement

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...

Homework Equations

Not sure.

The Attempt at a Solution

[/B]
Don't know where to start.

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?