Question

How does one solve the 2nd-order linear homogenous ODE?

Thanks for any help.

Answer 1

\[
\nabla ^2f(r)=0
\]Where:
\[
r=|\vec r|
\]

Answer 2

This can be re-written as
\[
\frac{d^2f}{dr^2}+\frac{2}{r}\frac{df}{dr}=0
\]

Answer 3

@Abhisar

Answer 4

For reference the general answer is:
\[
f(r)=\frac{c_1}{r}+c_2
\]

Answer 5

Oh, never mind, I'm just absolutely not right today.

Simple: distribute the \(r\), integrate by parts on the first term and you've got yourself a first-order linear ODE.