1. (a) (5 points) Use the Laplace transform method to solve the IVP y\" + 31/ + 2y = t26(t - 2), 31(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) + 2W) = W20), 11(0) = 2, 11(0) = -2- 2. A springmass system with damping, described by the equation 2:\" + 223' + 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 = 7r, and then it is struck with an impulse F6 at time t = T > 1r, where F # 0. Thus, the position w(t) of the mass obeys the symbolic IVP as\" + 233' + 2:: = 5(t 7r) + F60! T), $(0) = 0, 53(0) 2 0. (a) (5 points) Find the position 33(t) of the mass for all t 2 0. (b) (5 points) Given that T = 3n, nd the strength F such that 33(15) = 0 for all t 2 T = 37r, 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) = / \"26(3-0) cos(t v)dv. 0 (b) (6 points) Use Laplace transforms to solve the integrodifferential equation t y' = t +/ vy(t v)dv, y(0) = 0. 0