Q4. (Initial Value Problems - IVPs) Solve the following initial value problems (IVPs) by the Laplace transform.
(i) $y' + 5y = 10 \sin 2t$; $y(0) = 0$
(ii) $y'' + 16y = 2e^{-2t}$; $y(0) = 0$, $y'(0) = 0$
(iii) $y'' + 3y' + 2y = 0$; $y(0) = 3$, $y'(0) = -2$
(iv) $y'' + 4y' + 4y = 8t^3 + 12$; $y(0) = 1$, $y'(0) = 3$