Use LaPlace transformation to solve the following initial value problem.

Question

Express as a function of t.

y'' + 6y' + 5y = 24e^t

y(0) = 2

y'(0) = 6

Answer 1

L(y''+ 6y' +5y) =L(24e^t);

s²F(s) - 2s - 6 + 6(sF(s) - 2) + 5F(s) = 24/(s - 1);

F(s)(s² + 6s + 5) = 24/(s - 1) + 2s + 18;

F(s) = (2s² + 16s + 6)/[(s - 1)(s + 1)(s + 5)]. Partial fraction decomposition give us:

F(s) = 1/(s + 1) + 2/(s - 1) - 1/(s + 5).

Now we applaying inverece Laplace: L^-1(F(s)) = L^--19(1/(s + 1) + 2/(s - 1) - 1/(s + 5).);

y(t) = e^-t + 2e^t - e^-5t