Take the Laplace transform of the IVP $$\frac{d^2y}{dt^2}+k^2 y = e^{-6 t}, y(0) = 0, y^\prime(0)=0$$ Use $Y$ for the Laplace transform of $y$, (not $Y(s)$).



So

$Y =$

$\hskip 35pt 1$ -----------------------------

$\hskip 10pt =$

----------------------------------- $\hskip 32pt s + 6$

$+$

$s$$+$ -------------------------------------------------------------- $\hskip 70pt s^2+k^2$

and $y(t) =$