Question

Obtain the Laplace transform of the functions plotted in Figures 1 and 2.

Obtain the Laplace transform of the functions plotted in Figures 1 and 2. ft) f(t) 14 0 10 20 30 40 Figure 1 Figure 2

Answer 1

(i) For figure 1, f(t) = {2 0 lessthanorequalto t lessthanorequalto 2 -1 2 lessthanorequalto t lessthanorequalto 3 0 3 lessthanorequalto t lessthanorequalto infinity L{f(t) } = integral e^-st f(t) dt = integral _0 ^2 e^-5t (t) dt + integral_2 ^3 e^-5t (-1) + integral _3 ^infinity e^-5t (0) dt = 2[e^-5t/-5]_0^2 - [e^-5t/-5]_2 ^3 + 0 = -2 /5(e^-2s -1) + 1 /5 (e^-3s - e^-2s) = -3/5 e^-2s - 2/5 + 1/5 e^-3s Therefore F(t) = {f(t) } = 2/5(e^-3s - 3e^-2s) - 2/s F(s) = L{f(t) } = L/s(e^-3s - 3e^-2s) - 2/s Therefore L{f(x) } = 1/s (e^-3s 3e^-2s -2) (ii) For figure 2, f(t) = {14/10 t, 0 lessthanorequalto t lessthanorequalto 10 14, 10 lessthanorequalto t lessthanorequalto 30 14/10(40 - t) , 30 lessthanorequalto t lessthanorequalto 40 0, t greterthan 40 L{f(t) } = integral e^-st f(t) dt L{f(t) = > e^-st times 7/5 t dt + integral e^-5t times 14 dt + integral e^-5t times 7/5 (40-t) dt + integral e^-5t (0) dt