Solve using the laplace transform y 2y 6u1t 6u3t subject

Solve using the laplace transform y\' + 2y = 6u_1(t) - 6u_3(t) subject to y(0) = 6

Solution

Taking the Laplace Transform of the given equation

L[ y\' ] + 2L[ y ] = 6L[ u(t - 1) ] - 6L[ u(t - 3) ]

sY - y(0) + 2Y = (6e^-s/s) - (6e^-3s/s)

Plugging the given value of y(0) = 6

sY - 6 + 2Y = (6e^-s/s) - (6e^-3s/s)

Y(s + 2) = (6e^-s/s) - (6e^-3s/s) + 6

Y = ( 1/(s + 2) )( (6e^-s/s) - (6e^-3s/s) + 6 )

Taking Inverse Laplace Transform

y(t) = 6e^-2t + 3u(t - 1) - 3u(t - 3) - 3e^2-2tu(t - 1) + 3e^6-2tu(t - 3)

where u(t) denotes unitstep function