 Solution

Write $\omega =\sqrt{2}$ and $Y(p)=\overline{y}(p)$. The zero initial data and the Laplace transform of a derivative give $\mathcal{L}[y^{\prime \prime }]=p^{2}Y$. Transforming the forcing with part (b),ThereforePart (b) with $\omega =\sqrt{2}$ gives $\mathcal{L}[t^2\sin (\sqrt{2}t)]=2\sqrt{2}(3p^2-2)/(p^2+2{)}^3$. Comparing coefficients and using uniqueness of the Laplace transform,As a direct check, $y^{\prime }=\frac{t}{2}\sin (\sqrt{2}t)+\frac{\sqrt{2}{t}^2}{4}\cos (\sqrt{2}t)$, so both initial values vanish. Differentiating once more cancels the $t^2\sin (\sqrt{2}t)$ terms in $y^{\prime \prime }+2y$, leaving $\sqrt{2}t\cos (\sqrt{2}t)+\frac{1}{2}\sin (\sqrt{2}t)$ as required. This also checks the factor $1/4$ in the inverse transform.