Problem 2: (a) Use the definition of the Laplace transform and a change of variables to show that 1 T &z - Vi s You will need to use the following famous formula for the area under a Gaussian (bell curve): 2 J e dx =1/ -0 (Note: One place where you can learn a trick to calculate this definite integral, the area under the Gaussian, is Math 215). Vr (b) Deduce from part (a) that & {\\/;} B ; 2432 Problem 3: Consider the model of a pendulum that is struck with a hammer at timet =T (where T > (), transferring momentum A. This system can be modeled by Y'+2ry +wjy = As(t T) (a) Suppose we have y = 2 and @, = \\/g . Find the gquasi-period for the unforced oscillator. (b) Now assume the shock occurs when one quasi-period has passed (that is, T is the quasi- period from part (a)). Furthermore, assume that we have initial conditions y(0) = 0, and '(0) = 1. Solve the problem using Laplace transforms and determine the value of A that will result in the solution being identically zero forall > T. (c) Now assume the shock occurs instead when two quasi-periods have passed (that is, T is twice the guasi-period from part (a)). Again, assume that we have the initial conditions y(0) =0, and y'(0) = 1. Repeat your process from part (b) and find the value of A that will result in the solution being identically zero forall t = 7. (d) Compare your values of A from parts (b) and (c). Is one larger than the other? Do your results make sense physically? Problem 4: (a) Find the Laplace transform F(s) of the function () (b) Find the solution of the ODE f) = Y'+y = : r sm | 2 with initial conditions y(0) = y'(0) = 0. You may leave your answer as an infinite sum. Determine if the solution remains bounded as oo