Solving ∂²Φ/∂s² + (1/s)∂Φ/∂s - C = 0 for radial Φ

[Daniel Sellers]

TL;DR

I have a fairly simple non-homogeneous second order ODE to solve but I can't seem to get started with it.

I need to solve

∂²Φ/∂s² + (1/s)*∂Φ/ds - C = 0

Where s is a radial coordinate and C is a constant.

I know this is fairly simple but I haven't had to solve a problem like this in a long time. Can someone advise me on how to begin working towards a general solution?

Is the method of undetermined coefficients the correct approach?

Thanks very much.

 

[Infrared]

Substituting $\Psi=d\Phi/ds$ turns your equation into the first order equation $\Psi'(s)+\frac{1}{s}\Psi(s)=C$. You should be able to do this with an integrating factor.

 

[Chestermiller]

The equation is the same as $$\frac{1}{s}\frac{\partial}{\partial s}\left(s\frac{\partial \phi}{\partial s}\right)=C$$

 

[HallsofIvy]

Is there a reason for the partial derivative symbols when there is only the single variable s?