Problem 1. (40 points) A spring-mass system has a spring constant of 6 N/m. A mass of 4 kg is attached to the spring, and the motion takes place in a viscous uid that results in a damping coefcient equal to 2 kg/sec. An external force 6 cos 3t 4 sin 31? N is applied on the system. (1) Write down an ODE describing the displacement u(t) of the mass from its equilibrium position at time t. Then nd the general solutions of the ODE you found. (2) Given the initial displacement 11(0) = % m and the initial velocity u' (0) = % m/sec, nd the solution satisfying these initial conditions. (3) Is the solution you found in part (2) a periodic function? Find the amplitude and the frequency. Problem 2. (30 points) In each of the following problems, the polynomial Z (r) is the characteris- tic polynomial of a nth order homogeneous ODE with constant coefcients, and you're given the factorization of the Z (7") Find the general solutions of the corresponding ODE. (1) Z(T')=T'3T2+Tl=(T1)(T2+1) (2) 20") = r4 2T3 + 2T2 27' +1 = (r 1)2('r'2 + 1) (3) Z0") 2 (r 2)2(7'2 +7' +1)2. Problem 3. (30 points) Find the general solution of the ODE 31\"\" + 2y" + y = 0