Q7. (Learning Outcome 1) Use Laplace Transform method to solve the following initial-value problem: a) y" + 2y'ty= tet, y(0) = 0, y'(0) = 0 b) y" + 1 = cos (t), y(0) = 0, y'(0) = 0 c) Assume that an engineering system can either be in the three states: i = 0 (operating normally), i = 1 (failed due to hardware problem), or i = 2 (failed due to human errors). The following set of differential equations describes the probability (X;(t)) that the engineering system is in state i at time t (in hours) where i = 0, 1, 2: dxo (t) -+ 0.02X.(t) = 0 dt dX1(t) 0.015X (t) = 0 dt dX 2 (t) 0.005X. (t) = 0 dt At the initial time, we know that the engineering system is operating normally. Apply Laplace Transform to solve the engineering problem and and find out the mean time to system failure, MTTF = So Xo(t) dt