Assuming right form form homogeneous solution, exp(r t), which is recognizing it's linear, constant coefficient: 5 points
5 points for the right form for homogeneous solution (because of repeated roots)
10 points for right form for particular solution, A t + B (not doing B is a big mistake)
5 points for right algebra for undetermined coefficients and assembling into right answer
Problem 2:
+5 For recognizing the mechanics/how the bolted frame moves
+5 For recognizing the mechanics/how the spring & damper system moved (credit given for correct xp or differential equation if xp was never written)
+5 For a reasonable expression of fg (not quite solving for the spring or damping components, but close). This was also the part where points were added/deducted for a correct explanation of how the system would be affected by the spring/damping system. Points were deducted for not taking into account the applied spring and damping forces against the ground.
+5 For the correct spring component
+5 For correctly evaluating the damping component.
Problem 3:
5 points: Obtaining Lambda and getting general form of solution (real and complex)
10 points: Answer to (a). Showing that as t--> inf, limit goes to 0 in all cases (real and complex)
10 points: Answer to (b). All three cases (4 for any one case, 3 each for the other two cases)