Answer:
Y(s) = ((8)/(s^2-6s+27)(s^2+64)) + (6s-29)/(s^2-6s+27)
Step-by-step explanation:
Let's find the Laplace transform of each component of the differential equation.
Y'' = L{y''(t)} = s^2Y(s) - sy(0) - y'(0) = s^2Y(s) - 6s - 7
Y' = L{-6y'(t)} = -6L{y'(t)} = -6(sY(s) - y(0)) = -6sY(s) +36
Y = L{27y(t)} = 27L{y(t)} = 27Y(s)
L{sin(at)} = (a)/(s^2 + a^2), so
L{sin(8t)} = 8/(s^2 + 64)
Now, putting all of this into the original differential equation, we have:
s^2Y(s) - 6s - 7 - 6sY(s) + 36 + 27Y(s) = 8/(s^2+64)
Y(s)(s^2-6s+27) = (8/(s^2+64)) + 6s - 29
Solving for Y(s), we end up with:
Y(s) = ((8)/(s^2-6s+27)(s^2+64)) + (6s-29)/(s^2-6s+27)
