Question

Solve the IVP:

Answer 1

Given IVP y^11 + 2y^1 + 3y = sin t + delta (t - 3 pi): y(0) = 0, y^1(0) = 0 applying laplace transform on both sides L[y^11] + 2 L[y^1] + 3 L[y] = L(sin t) + L(delta (t - 3 pi)] we know that L(y^11) = s^2 y(s) - s y(0) - y^1(0) L(y61) = s y(s) - y(0) where L [y(t)] = y(s) L[sin at] = a/s^2 + a^2 L[s (t - c)] = e^-cs therefore s^2 y(s) - s y(0) - y^1(0) + 2 (s y(s) - y(0)) + 3 y(s) = 1/s^2 + 1 + e^-3 pi s s^2 y(s) - s(0) - (0) + 2 s y(s) - 0 + 3 y(s) = 1/s^2 + 1 + e^-3 pi s y(s) (s^2 + 2s + 3) = 1/s^2 + 1 + e^-3 pi s y(s) = 1/(s^2 + 2s + 3) (s^2 + 1) + e^-3 pi s /s^2 + 2s + 3 L[s(t)] = 1/(S^2 + 1) (s^2 + 2s+ 3)