I'm trying to understand a solved problem in our notes and can't find a similar technique used elsewhere, so I'm asking for some assistance in understanding what exactly they did.

The problem:

Find a general solution to the equation

and the particular solution when

Solution:

The characteristic equations are:

I don't really understand these next steps they do which is:

These imply that for some constant

and

from which it follows that two integrals are for constants ( )

or

There are a few more steps after this, but I guess I'll ask about those if I don't understand them after I understand these steps.

Thanks for any assistance.