Question

Hw9: Problem 12 Previous Problem Problem List Next Problem (1 point) In this exercise we will use the Laplace transform to solve the following initial value problem: 0<, î < 1· 3, y@) :0 (1) First, using Y for the Laplace transform of y(t), ie, Y = L(y(t), find the equation obtained by taking the Laplace transform of the initial value problem (2) Next solve for Y - (3) Finally apply the inverse Laplace transform to find y(t) y(t) =

Answer 1

first find the differential equation:

First find the differential equation: y'(t) + 3y(t) = -3(u(t) - u(t - 1)) y'(t) + 3y(t) = -3u(t) + 3u(t - 1) Apply the Laplace transformation both sides: L_i [dy(t)/dt + 3 y(t)] (s) = L_i [- 3u(t) + 3u(t - 1)] (s) L_i [dy(t)/dt + 3y(t)] (s) = L_i [- 3u'(t) + 3u(t - 1)] (s) L_i [d y(t)/dt](s) + 3 (L_i [y(t)] (s)) = -3 (L_i [u(t)] (s)) + 3 (L_i [u(t - 1)] (s)) s (L_i [y(t)] (s)) - y(0) + 3 (L_i [y(t)] (s)) = -3/s + 3e^- s/s Substitute: y(0) = 0 (s + 3) (L_i [y(t)] (s)) = - - 3 e^- s + 3/s