Given the second order initial value problem Let y 4y 4delt

Given the second order initial value problem Let y\" -4y = 4delta(t-3),y(0) = 1, y\'(0) = 1, y\'(0) = 10 Let Y(s) denote the Laplace transform of y. then Y(s) = Taking the inverse Laplace transform we obtain y(t) =

Solution

Let L[y] = Y

=> L[y\'\'] - 4L[y] = 4L[ u(t - 3) ]

=> s²Y - s.y(0) - y\'(0) - 4Y = 4e^-3s/s

=> s²Y - 4Y - s - 10 = 4e^-3s/s

=> Y = 4e^-3s/( s(s² - 4) ) + ( s + 10)/( s² - 4 )

Calculating Inverse Laplace Transform

y(t) = 0.5 ( e^2t-6u(t - 3) - 2u(t - 3) + e^-2t+6u(t - 3) + 6e^2t - 4e^-2t )