1. (a) (2 points) Evaluate $\int_{0}^{\infty} e^{-st}u_a(t)\delta(t-t_0)dt$ if (i) $0 \leq a < t_0$; (ii) $t_0 < a$.
(b) (4 points) Use Laplace transform to solve the IVP
$y'' + 4y' + 3y = t\delta(t-2)$, $y(0) = 1$, $y'(0) = 1$.
(c) (4 points) Solve the same initial value problem except this time replace the right-hand side of the differential equation by $tu_2(t)$; that is, solve
$y''(t) + 4y'(t) + 3y(t) = tu_2(t)$, $y(0) = 1$, $y'(0) = 1$.
