The following property of Laplace Transform (LP) is needed:
The first step is to take LP of evey term of the DE
What we did so far is LP of the solution, y(t). To find solution itself we have to take Inverse LP from RHS of (LP), which is
y(t)=-(17/10)-3 t+(13/5)e-2 t + (1/10)(cos(t)+sin(t))
I just give the answer, I suppose you have to do calculations yourself.