Question

1. y(3)-2y"+Sy.-0, y(0)-O, V00)-Ly( )-i using Laplace transform, solve y(t) and y"(0) af ecOS

Answer 1

"y""' - 2y"" + 5y = 0 y(0) = 1 y'(0) = 1 y(pi/8) = 1 Apply the L > T on both side ineqaution (1) L(y""' - 2y"" + 5y') = 0 S^3 Y(s) - s^2y(0) - s^1y(0) - y""(0) - 2[s^2y(s) - sy(0) - y'(0)] + 5{sy(s) - y(0)] = 0 (s^3 - 2s^2 + 5s) y(s) - s^2(0) - s(1) - y'(0) - 2(- s(0) - 1) + 5(- 0) = 0 (s^3 - 2s^2 + 5s) y(s) - s - y""(0) + 2 = 0 y(s)s(s^2 - 2s + 5) = s - 2 + y""(0) y(s) = s/s(s^2 - 2s + 5) - 2/s(s^2 - 2s + 5) + y""(0)1/s(s^2 - 2s + s) Y(s) = 1/s^2 - 2s + 5 - 2/s(s^2 - 2s + 5) + y""(0) 1/s(s^2 - 2s + 5 Y(s) = s - 2/s(s^2 - 2s + 5) + y""(0)1/s(s^2 - 2s + 5) s - 2/s(s^2 - 2s + 5) = A/5 + Bs + c/s^2 - 2s + 5"