Assuming you start with the homogeneous ODE,
upon taking the Laplace transform of both sides, you end up with
since the transform operator is linear, and the transform of 0 is 0.
I'll denote the Laplace transform of a function
into the
-domain by
.
Given the derivative of
, its Laplace transform can be found easily from the definition of the transform itself:
Integrate by parts, setting
so that
The second term is just the transform of the original function, while the first term reduces to
since
as
, and
as
. So we have a rule for transforming the first derivative, and by the same process we can generalize it to any order provided that we're given the value of all the preceeding derivatives at
.
The general rule gives us
and so our ODE becomes
Depending on how you learned about finding inverse transforms, you should either be comfortable with cross-referencing a table and do some pattern-matching, or be able to set up and compute an appropriate contour integral. The former approach seems to be more common, so I'll stick to that.
Recall that
and that given a function
with transform
, the shifted transform
corresponds to the function
.
We have
and so the inverse transform for our ODE is
and in case you're not familiar with hyperbolic functions, you have