Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

Question

$ty''(t) + y'(t) -ty(t)= tf(t)$

How to solve the problem using Laplace Transform?

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Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where $C$ is constant. What is the inverse?