1. (a) (5 points) Use the Laplace transform method to solve the IVP y\" + 3y' + 221 = 15260? - 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) = M203). That is, solve y\"(t) + 33%) + 2W) = W205), 9(0) = 2, y'(0) = -2- 2. A spring-mass system with damping, described by the equation :5\" + 23:" + 23: = 0, is initially at rest but the mass is struck twice with a hammer: First it is struck with a unit impulse 6 at time t = 71', and then it is struck with an impulse F6 at time t = T > 7r, Where F 75 0. Thus, the position a:(t) of the mass obeys the symbolic IVP 11:\" + 23' + 22: = 5(t 17) + F6(t T), 33(0) = 0, (3(0) = 0. (a) (5 points) Find the position $(t) of the mass for all t 2 0. (b) (5 points) Given that T = 371', nd the strength F such that x(t) = 0 for all t 2 T = 311', i.e., the second hammer strike perfectly cancels out the motion caused by the rst hammer strike. t 3. (a) (4 points) Find the Laplace transform of f (t) = f 2:2603\") cos(t v)dv. o (b) (6 points) Use Laplace transforms to solve the integrodifferential equation t y' = t +] vy(t v)dv, y(0) = 0. 0