(a) Take the Laplace transform of both sides:
where the transform of comes from
This yields the linear ODE,
Divides both sides by :
Find the integrating factor:
Multiply both sides of the ODE by :
The left side condenses into the derivative of a product:
Integrate both sides and solve for :
(b) Taking the inverse transform of both sides gives
I don't know whether the remaining inverse transform can be resolved, but using the principle of superposition, we know that is one solution to the original ODE.
Substitute these into the ODE to see everything checks out: