3.5 Find the Laplace transform of the following time functions (* denotes convolution): (a) f(t) = sin t sin 3t (b) f(t) = sin^2 t + 3 cos^2 t (c) f(t) = (sin t)/t (d) f(t) = sin t * sin t (e) f(t) = ??? cos(t - ?) sin ? d? 3.6 Given the Laplace transform of f(t) is F(s), find the Laplace transform of the following: (a) g(t) = f'(t) cos t (b) g(t) = ??? ???¹ f(?) d? dt? 3.7 Find the time function corresponding to each of the following Laplace transforms using partial-fraction expansions: (a) F(s) = 1 / (s(s + 2)) (b) F(s) = 10 / (s(s + 2)(s + 10)) (c) F(s) = (3s + 2) / (s^2 + 2s + 20) (d) F(s) = (3s^2 + 9s + 12) / ((s + 1)(s^2 + 5s + 11)) (e) F(s) = 2 / (s^2 + 4) (f) F(s) = (2(s + 2)) / ((s + 1)(s^2 + 4)) (g) F(s) = (s + 1) / s^2 (h) F(s) = 1 / s^6 (i) F(s) = 4 / (s^4 + 4)
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Find the Laplace transform of f(t) = sin(t) / sin(3t) To find the Laplace transform of this function, we can use the formula: L{f(t)} = ∫(e^(-st) * f(t)) dt, from 0 to ∞ L{sin(t) / sin(3t)} = ∫(e^(-st) * (sin(t) / sin(3t))) dt, from 0 to ∞ This integral is not Show more…
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