10:48 47 Aa 4» Q A 294 CHAPTER 7 THE LAPLACE TRANSFORM 41. y" + y = V2 sin V2r, y(0) = 10, y'(0) = 0 0.9s 21. 9 |(s - 0.1)(s + 0.2) 42. y" + 9y = e', y(0) = 0, y'(0) = 0 s- 3 43. 2y" + 3y" - 3y' - 2y = e, y(0) = 0, y'(0) = 0, y"(0) = 1 22. L V3)(s + V3)| 44. y" + 2y" - y' 2y = sin 3t, y(0) = 0, y'(0) = 0, y"(0) = 1 23. L-1 The inverse forms of the results in Problem 49 in Exercises 7.1 are (s - 2)(s - 3)(s - 6) 5? + 1 24. L = e"cos bt s(s - 1)(s + 1)(s 2) a)? + b? 25. L 26. L = sin bt. L³ + 5s] (s + 2)(s + 4) (s - a)? + b? 2s - 4 |(s + s)(s? + 1) In Problems 45 and 46 use the Laplace transform and these inverses to solve the given initial-value problem. 27. L-1 28. L 6я + 3 45. y' +y = e cos 21, y(0) = 0 29. L 30. L- (s2 + 1)(s + 4) 4+ 52 + 4 46. y" - 2y' + 5y = 0. y(0) = 1. y'(0) = 3 In Problems 47 and 48 use one of the inverse Laplace transforms In Problems 31-34 find the given inverse Laplace transform by finding the Laplace transform of the indicated function f. found in Problems 31-34 to solve the given initial-value problem. 47. y" + 4y = 10 cos 5t, y(0) = 0, y'(0) = 0 31. L f) - e" sinh bt (s - a) - b2 48. y" + 2y = 4t, y(0) = 0, y'(0) - 0 32. L s + a)) f) - at - sinat Discussion Problems 49. (a) With a slight change in notation the transform in (6) is the same as 33. 9- T+a®%& + b°* f(1) = a sin bt - bsinat L(f'()) = sL(fO) - f(0). With f(t) = te", discuss how this result in conjunction with (c) of Theorem 7.1.1 can be used to evaluate £{te"}. 34. L (s + a'Ms? + h ) = cos bt - cos at (b) Proceed as in part (a), but this time discuss how to use (7) with f(t) = t sin kt in conjunction with (d) and (e) of Theorem 7.1.1 to evaluate (t sin kt). 7.2.2 Transforms of Derivatives In Problems 35-44 use the Laplace transform to solve the given initial-value problem. 50. Make up two functions fi and f2 that have the same Laplace transform. Do not think profound thoughts. dy 35. - y = 1, y(0) = 0 51. Reread (iii) in the Remarks on page 293. Find the zero-input dt and the zero-state response for the IVP in Problem 40. dy + y = 0, y(0) = -3 di 36. 2 52. Suppose f() is a function for which f"(1) is piecewise continuous and of exponential order c. Use results in this Section 7.1 to justify 37. у' + буe", у(0) - 2 section 38. v' - y = 2 cos 5t, v(0) = 0 39. " + 5v' + 4y = 0, v(0) = 1, v'(0) = 0 f (0) = lim sF(s), %3D 40. " - 4y' = 6e" – 3e, v(0) = 1, y'(0) = -1 where F(s) = L(fO). Verify this result with f(t) = cos kt. 7.3 Operational Properties I INTRODUCTION It is not convenient to use Definition 7.1.1 each time we wish to find the Laplace transform of a function f(t). For example, the integration by parts involved in evaluating, say, L{e'f sin 3t} is formidable, to say the least. In this section and the next we present several labor-saving operational properties of the Laplace transform Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-202 294 Reader Contents Notebook Bookmarks Flashcards IK ШО