Output of system defined by differential equation
What you want to do is take the Laplace transform of both sides of S.  The first and second derivative terms will be replaced by expressions of $Y(s)$ and the conditions at $t=0-$, so you can solve the equation for $Y(s)$ and then do the inverse Laplace transform, or more commonly, use partial fraction expansion to go back to the time domain.  

See numbers 35 and 36 in the table of Laplace transforms here.