# Solving a non-homogeneous ODE with Bessel functions?

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
hh = c1J0(λr) + c2Y0(λr),
where J0 and Y0 are Bessel functions.

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

3. The Attempt at a Solution

To find hp, 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!

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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
hh = c1J0(λr) + c2Y0(λr),
where J0 and Y0 are Bessel functions.

Now I was planning on using h(r) = hh + hp,
where hp is a particular solution of the ODE.
A simple particular solution is easily found by inspection, and it is given by $$h_p=1/{\lambda}^2$$.