Question

6. Problem 6 Find the solutions in the time domain of the following second-order differential equation using the Laplace transform, (7a) (7b) (7c) y(0)-1; (0) =-1.

Answer 1

y^''(t) + y(t) = t -(1) given that y(0) = 1, y^'(0) = -1 Taking laplace transform on both side L{y^''(t)} + L{y(t)} = L{t} S^2 L{y} - S y(0) - y^'(0) + L{y(t)} = 1/S^2 S^2 L{y} - S Times 1 - (-1) + L{y} = 1/S^2 S^2 L{y} + L{y} - S + 1 = 1/S^2 S^2 + 1) L{y} = 1/S^2 + S - 1 (S^2 + 1) L{y} = 1 + S^3 - S^2/S^2 L{y} = 1 + S^3 - S^2/S^2 (S^2 + 1) = 1/S^2 + S - 2/(S^2 + 1) L{y(t)} = 1/S^2 + S/S^2 + 1 - 2/S^2 + 1 Taking Laplace inverse in both side y(t) = L^-1{1/S^2} + L^-1{S/S^2 + 1} - 2L^-1{1/S^2 + 1} = L^-1{1/S^2} + L^-1{S/S^2 + 1} - 2 L^-1{1/S^2 + 1} y(t) = t + cost - 2 sint Ans