Use the Laplace Method to solve the following IVP

Question

Use the Laplace Method to solve the following IVP:

                                          

Compute     and its Inverse Laplace Transform 

Answer 1

*Table Of Laplace Transforms

Second Order Linear ODE

Apply L( ODE )

L {x''(T)} = s^2 F(s) - s f(0) - f'(0) = s^2 F(s) -3*s

L {x(T)} = 4*L x(s)

L {(t-2)} = (1 - 2s) / s^2

L {u(t-2)} = (e^-2s)/s

L {d(t-1)} = e^-s

F(s) [s^2 +4] = 3s+e^-s+ [ e^-2s (1-2s)]/(s^3)

F(s) = ( 3s+e^-s+ [ e^-2s (1-2s)]/(s^3)) / (s^2 +4)

F(s) = [3e^s*s^4 + s^3 + e^-s(1-2s)] / [e^s*s^3(s^2+4)]

L^-1 (F(s) = [3e^s*s^4 + s^3 + e^-s(1-2s)] / [e^s*s^3(s^2+4)])

y(T) = 1/16 [H(t-2)(2t^2 -16t -4sin(4-2t)+cos(4-2t)+23)+8(H(t-1)sin(2t-2)+6cos(2t)]

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-Best Maksim