Question

5. STEP FUNCTIONS/ LAPLACE TRANSFORM g(t) 2 a) Express in terms of HEAVISIDE STEP FUNCTIONS b) Find the Laplace Transform using part (a)-show work very carefully and state any Theorems about Laplace Transforms you are using c) Find the Laplace transform by direct integration and not using any step functions. Show steps of integration work very carefully

Answer 1

(a) From above we get g (t) = t [u (t - 1) - u (t - 2)] + 1 . u (t - 2) (b) or, g (t) = (t - 1 + 1) [u (t - 1) - u (t - 2)] + u (t - 2) or, g (t) = u (t - 1) - u (t - 2) + (t - 1) u (t - 1) - (t - 2) u (t - 2) Taking Laplace transform on both sides, L {g (t)} = L {u (t - i)} - L {u (t - 2)} + L {(t - 1) u (t - 1)} - L {(t - 2) u (t - 2)} = 1/s e^- s - 1/s e^- 2 + e^- s . 1/s^2 - s^- 2 s . 1/s^2 [Because L {u (t - a)} = 1/s e^- a s and L {f (t - a) H (t - a)} = e^-as L {f (t)}] Therefore L {g (t)} = 1/s^2 (s e^- s - s e^- 2 s + e^- s - e^- 2 s) \r\n (e) Also from graph, g (t) = {0, when 0 lessthanorequalto t < 1 t, when 1 lessthanorequalto t < 2 1, when t greaterthanorequalto 2 Then by definition of Laplace transform, L {g (t)} = integral^infinity_0 e^- s t g (t) dt = integral^2_1 t e^- st dt + integral^infinity_2 1 . e^- st dt [using definition of g (t)] = [t . e^- st/(- s)]^2_1 - integral^2_1 1 . e^-s t/(- s) dt