La Place Transformations

Table of La Place Transformations for reference

8.1 Exercises 1. Find the Laplace transforms of the following functions by evaluating the integral F(s) = So estf(t) dt. (at (b) te-t (c) sinh bt (d) e2t - 3et (e) t2 2. Use the table of Laplace transforms to find the Laplace transforms of the following functions. (a) cosh t sin t (b) sin' t (c) cos2 2t (d) cosh' t (e) t sinh 2t (f) sin t cost (g) sin (t+ (h) cos 2t - cos 3t (i) sin 2t + cos 4tSection 8.1: Section 8.1 Homework: 1) Work through this worksheet first to practice finding the Laplace transforms of the following functions. Some include applying the "First Shift Theorem" or use the appropriate rules on the Table of Laplace Transforms 2) Complete the following problems from Section 8.1 ebook: la, 2a, 2b, 2c, 2d, 2e, 2f, 2h, 2i Find the Laplace transform for the following functions. 1) 2e-3 + 4 2) 2+3 - 312 + 5t - 2 3) 5e- sin(3t) 4) el cos(2t) + 4e' sin(2t) 5 ) est ( t + 5 ) 6) 7the 8t 5 7) -4t + 8) 3e- - 4te-tSection 8.1 Homework Solutions (and handout] Show all appropriate work in your solution, including writing the constant outside of the ${f(t)} symbol before using the table You can leave your answer in \"factored\" form and do not have to find common denominators, but I provided both forms as applicable. Handout: 2 4 6s+12 + OH 5+3 3 s(s+3) 1) 4) 5+7 (3 1)2+4 5314 5' (s 3)2 14 (s 8)3 5 6 11.9 + 169 + OH 6(5 + 4) 5(s 1) 30(s 1)(s + 4) 5) 7) 3 4 35 l OR. (5+1) (5+1)2 (5+1)2 8) From Section 8.1 ebook: 1 1a) Use the definition of the Laplace Transform to solve (improper integral) F (s) = 2 s 1 1 52 + 2 2a) + on 2[(s1)2+l] 2[(s+l)2+ l] [(s l)2+1][(s+1)2+1] l s 2 2b) _ , OR _ 23 2(52 + 4) 3(33 + 4) l 5 5248 2c} + OR+ 23 2(52 +16) 3(52 +16! 1 1 1 522 2d) + + + on a 4(5 2) 23 4(s + 2) s(s~ 4) l l 43' ZGWOR }2(5F2)2 2(s+2)2 (324):l 2f) 1 32 + 4 5 s 55 2h) _ OR .92 + 4 5'2 + 9 ('53 + 4N5: + 9) 2 s s3+252+4s+32 + OH 2' 033+4 s2+16 (52+4)(.92+16) Table of Laplace Transforms Notation: Sf{f(t)}, 39000)) 1 Notations for \"Transform Pair\": Example: 1 s