PROBLEM 1 (25 points): (a) (5 points) Consider the differential equation below: day dt 2 + 6 y dt + 5y = u Given u = e-2tu(t), find the output y using the Laplace transform and inverse Laplace transform only. No marks will be given for answers that do not use the Laplace transform. Assume initial conditions y(0), y'(0) = 0. In one sentence, justify why you used unilateral or bilateral Laplace transform. (b) (5 points) Find the regions of convergence of the (two sided) inverse Laplace transform of W = - S $ 2 - 9 Recall that there can be multiple functions, each with a different region of convergence, that correspond to the same Laplace transform. Hint: the boundaries of the regions of convergence correspond to special values of Re(s) in the Laplace transform (c) (10 points) Find the inverse Laplace transform of S W = $2 - 9 for each region of convergence that you obtained in (b). (d) (5 points) Explain briefly why the region of convergence needs to be specified for the bilateral Laplace transform but not the unilateral Laplace transform