1. (a) (5 points) Use the Laplace transform method to solve the IVP y" + 3y' + 2y = 128(t -2), y(0) = 2, y'(0) = -2. (b) (5 points) Solve the same initial value problem except this time replace the right-hand side of the differential equation by g(t) = tuz(t). That is, solve y" (t) + 3y' (t) + 2y(t) = tuz(t), y(0) = 2, y'(0) = -2. 2. A spring-mass system with damping, described by the equation a" + 2x' + 2x =0, is initially at rest but the mass is struck twice with a hammer: First it is struck with a unit impulse o at time t = 7, and then it is struck with an impulse Fo at time t = T > 7, where F * 0. Thus, the position x(t) of the mass obeys the symbolic IVP x" + 2x' + 2x = 8(t - 7) + FS(t -T), x(0)=0, x'(0)=0. (a) (5 points) Find the position x(t) of the mass for all t 2 0. (b) (5 points) Given that T = 37, find the strength F such that x(t) = 0 for all t 2 T = 37, i.e., the second hammer strike perfectly cancels out the motion caused by the first hammer strike. 3. (a) (4 points) Find the Laplace transform of f(t) = / ve (t-v) cos(t - v)dv. (b) (6 points) Use Laplace transforms to solve the integro-differential equation y' = t+ / vy(t - v)du, y(0) = 0