Solving a non-homogeneous ODE with Bessel functions?

[TheJCBand]

Hi, I posted this on the homework forum, but I haven't gotten any responses there. I thought there might be a better chance here.

1. Homework Statement

I have the ODE
h'' + h'/r + λ²h = 1,
where h = h(r), and I want to find h(r).

2. Homework Equations

The corresponding homogeneous equation is a Bessel equation that has the solution
h_h = c₁J₀(λr) + c₂Y₀(λr),
where J₀ and Y₀ are Bessel functions.

Now I was planning on using h(r) = h_h + h_p,
where h_p is a particular solution of the ODE.

3. The Attempt at a Solution

To find h_p, I tried using variation of parameters, but I get to a point where I need to both differentiate and integrate a Bessel function, which turns out to be pretty hard. I'm wondering if I'm going in the wrong direction, or if my logic here is even valid.

Thanks!

 

[Old Smuggler]

Hi, I posted this on the homework forum, but I haven't gotten any responses there. I thought there might be a better chance here.

1. Homework Statement

I have the ODE
h'' + h'/r + λ²h = 1,
where h = h(r), and I want to find h(r).

2. Homework Equations

The corresponding homogeneous equation is a Bessel equation that has the solution
h_h = c₁J₀(λr) + c₂Y₀(λr),
where J₀ and Y₀ are Bessel functions.

Now I was planning on using h(r) = h_h + h_p,
where h_p is a particular solution of the ODE.

A simple particular solution is easily found by inspection, and it is given by $$h_p=1/{\lambda}^2$$.