What do you mean "trial solutions"?
I'll try to find a solution of 1)... but I don't know if it is or not the 'trial solution'...
Using t as independent variable the DE is...
(1)
If is the Laplace Tranform of f(t), then a basic property is...
(2)
Now if we use the property (2) in (1) we obtain...
(3)
... that with some steps becomes...
(4)
... so that we have now a first order linear DE in s, of course more 'approachable' then (1). The (4) can be solved with 'standard procedure' obtaining...
(5)
At this point if You have a look at the manual, You discover that is...
(6)
... where is the Bessel function of first kind of order 0. Only one consideation: in (6) we have only one 'arbitrary constant' c even if the (1) is a linear ODE of order two. The [probable] reason is that the procedure finds only the solutions of (1) that are L-transformable, so that the other independent solution, probably with a singularity in t=0, is 'desaparecida'...
Kind regards
Now we pass to...
(1)
... which is a little modified respect to the ODE 2) in the original post [1 instead of 3...]. Here we only want to show how to proceed in cases like this...
In order to eliminate the 'unconfortable' term , let's suppose that is . In thast case is...
(2)
(3)
... and, inserting (3) in (1), we have...
(4)
The (4) is identified as a Bessel differential equation of order and its solution is...
(5)
... so that the solution of (1) is...
(6)
Kind regards