Solve the IVP of the following question.

Question

Solve the following IVP:

 

Answer 1

We revrite right side : x''(t) - x(t) = (t - 2)u₂(t) + (3 - t)u₅(t) = (t - 2)u₂(t) - ((t - 5) + 2))u₅(t).

Then laplace Transform give us: s²F(s) - F(s) = e^-2s·1/s² - e^-5s(1/s² + 2/s).

From here F(s) = e^-2s(1/(s²(s² - 1)) - e^-5s((2s + 1)/(s²(s² - 1)). After performing partial fraction decomposition we get:

F(s) = e^-2s(- 1/s² + 1/2·1/(s - 1) - 1/2·1/(s + 1)) - e^-5s(- 1/s² - 2/s - 3/2·1/(s - 1) + 1/2·1/(s + 1)). Then inverse Laplace.

x(t) = u₂(t)[- (t - 2) + 1/2e^{t - 2} - 1/2e^{- t + 2}] + u₅(t)[- (t - 5) - 2 - 3/2e^{t - 5} + 1/2e^{- t + 5}]

x(t) = u₂(t)[- t + 2 - 1/2e^{t - 2} 1/2e^{- t + 2}] + u₅(t)[- t + 3 - 3/2e^{t - 5} + 1/2e^{- t + 5}]