1. (a) (5 points) Use the Laplace transform method to solve the IVP y" + 3;; + 2;; = rst 2). m0) = 2. m) = 2- (b) (5 points) Solve the same initial value problem except this time replace the right-hand side of the dierential equation by g(t) = tag(t). That is, solve 9%) + 31/0) + 2W) = $112\"): 9(0) = 2, #(0) = -2' 2. A spring-mass system with damping. described by the equation 9:\" + 292' + 2:: = 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 = 11', and then it is struck with an impulse F6 at time t = T > 1r, where F 96 0. Thus, the position a:(t) of the mass obeys the symbolic IVP 0, 2"(0) = 0. 2:" + 233' + 23 = 6(t r) + F5(t T). 3(0) (a) (5 points) Find the position :e(t) of the mass for all t 2 0. (b) (5 points) Given that T = 3s, nd the strength F such that z(t) = D for all t 2 T = 3s, i.e., the second hammer strike perfectly cancels out the motion caused by the rst hammer strike. c 3. (a) (4 points) Find the Laplace transform of f(t) = f v2e"(""'} cos(t skin. a (b) (6 points) Use Laplace transforms to solve the integro-di'erential equation 1'. y' = l + f vy as. me) = o. 0