Problem 1 (9pt) In this question, we will find the Laplace transform of cos wt in...

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Problem 1 (9pt) In this question, we will find the Laplace transform of cos wt in three different ways. (a) Follow the three steps to find the Laplace transform of cos wt from scratch. • Show Cletat = L[cos wt] + iL[sin wt], (1) from efwt = cos wt + i sinwt and the linearity property. • Show Lļet] = +i- 82 + w? (2) using Lleat] = for any complex number a. • Using that fact that (1) and (2) are identical, show L[cos wt) : %3D s2 + w? (b) Let f(t) = sin wt. We have F(s): (3) s2 + w?" • Show cos wt = (4) • Find the Laplace transform of (4) Using L[f'(t)) = sF(s) – f(0) and (3). (c) Let g(t) = cos wt. • Show g"(t) = -w? cos wt. (5) • Take the Laplace transform of (5) to show s°G(s) – s = -w*G(s). - 8 (6) • Solve (6) for G(s). Problem 1 (9pt) In this question, we will find the Laplace transform of cos wt in three different ways. (a) Follow the three steps to find the Laplace transform of cos wt from scratch. • Show Cletat = L[cos wt] + iL[sin wt], (1) from efwt = cos wt + i sinwt and the linearity property. • Show Lļet] = +i- 82 + w? (2) using Lleat] = for any complex number a. • Using that fact that (1) and (2) are identical, show L[cos wt) : %3D s2 + w? (b) Let f(t) = sin wt. We have F(s): (3) s2 + w?" • Show cos wt = (4) • Find the Laplace transform of (4) Using L[f'(t)) = sF(s) – f(0) and (3). (c) Let g(t) = cos wt. • Show g"(t) = -w? cos wt. (5) • Take the Laplace transform of (5) to show s°G(s) – s = -w*G(s). - 8 (6) • Solve (6) for G(s).

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a e iwt COS wt i sinwt Applying Laplace tsanstoam ... View the full answer