1. (a) (5 points) Use the Laplace transform method to solve the IVP y\" + 31/ + 2y = t25(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) : tu2(t). That is, solve y\"(t)+3y'(t)+2y(t)=tu2(t), y(0)=2, y'(0)=2. 2. A springmass system with damping, described by the equation :6\" + 21:" + 23: = O, 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 F 5 at time t = T > 1r, where F a 0. Thus, the position $(t) of the mass obeys the symbolic IVP x\" + 233' + 2x = 5(t 1r) + F603 T), 33(0) = 0, as'(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 $(t) = O for all t 2 T = 311', La, the second hammer strike perfectly cancels out the motion caused by the rst hammer strike. 1 3. (a) (4 points) Find the Laplace transform of f (t) = ] Uze(tU) cos(t v)d'u. 0 (b) (6 points) Use Laplace transforms to solve the integrodifferential equation 0. t y' = t +f vy(t v)dv, 9(0) 0