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y' - Sy=e-2¹, y(0) = 1

y' - Sy=e-2¹, y(0) = 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 19 please!

8. y(t) = 1² 10. y(t) = e²¹ 12. y(t) = cos 3t In a manner similar to that proposed in Exercises 8-13, verify the result of Proposition 2.4 for the functions defined in Exer- cises 14-17. 14. y(t) = t4 16. y(t) = sin 2t In Exercises 18-25, use Propositions 2.1, 2.4, and 2.7 to trans- form the given initial value problem into an algebraic equation involving (y). Solve the resulting equation for the Laplace transform of y. 9. y(t) = t³ 11. y(t) = e 3t 13. y(t) = sin 5t 21. y' - 4y = 18. y' + 3y = t², y(0) = -1 19. y' - 5y = e-2¹, y(0) = 1 20. y' + 5y = t² + 2t + 3, y(0) = 0 15. y(t) = e 2t 17. y(t) = t² + 3t+5 cos 2t, y(0) = -2 22. y" + y = sin 4t, y(0) = 0, y'(0) = 1 23. y" + 2y' + 2y = cos 2t, y(0) = 1, y'(0) = 0 24. y"+y' + 2y = cos 2t + sin 3t, y(0) = -1, y'(0) = 1 -t 25. y" + 3y' + 5y = t + e¹, y(0) = -1, y'(0) = 0 26. y(t) = et sin 3t 28. y(t) = e-2¹ (2t + 3) In Exercises 26-29, use Proposition 2.12 to find the Laplace transform of the given function. 30. y(t) = t sin 3t 32. y(t) = t² cos 2t 27. y(t) = e²t cos 2t 29. y(t) = e-t (t² + 3t+4) In Exercises 30-33, use Proposition 2.14 to find the Laplace transform of the given function. 31. y(t) = te-t 33. y(t) = t²e²t In Exercises 34-41, use the propositions in Section 2 to trans- form the given initial value problem into an algebraic equation transform of y. involving L(y). Solve the resulting equation for the Laplace 34. y' + 2y = t sint, y(0) = 1 35. y' - y = 1²e-², y (0) = 0 36. y' + y = e' sin 3t, y(0) = 0 37. y - 2y = e²¹ cost, y(0) = -2 38. y" + 4y = f² sin 4r, y (0) = 0, y' (0) = -1
Transcribed Image Text:8. y(t) = 1² 10. y(t) = e²¹ 12. y(t) = cos 3t In a manner similar to that proposed in Exercises 8-13, verify the result of Proposition 2.4 for the functions defined in Exer- cises 14-17. 14. y(t) = t4 16. y(t) = sin 2t In Exercises 18-25, use Propositions 2.1, 2.4, and 2.7 to trans- form the given initial value problem into an algebraic equation involving (y). Solve the resulting equation for the Laplace transform of y. 9. y(t) = t³ 11. y(t) = e 3t 13. y(t) = sin 5t 21. y' - 4y = 18. y' + 3y = t², y(0) = -1 19. y' - 5y = e-2¹, y(0) = 1 20. y' + 5y = t² + 2t + 3, y(0) = 0 15. y(t) = e 2t 17. y(t) = t² + 3t+5 cos 2t, y(0) = -2 22. y" + y = sin 4t, y(0) = 0, y'(0) = 1 23. y" + 2y' + 2y = cos 2t, y(0) = 1, y'(0) = 0 24. y"+y' + 2y = cos 2t + sin 3t, y(0) = -1, y'(0) = 1 -t 25. y" + 3y' + 5y = t + e¹, y(0) = -1, y'(0) = 0 26. y(t) = et sin 3t 28. y(t) = e-2¹ (2t + 3) In Exercises 26-29, use Proposition 2.12 to find the Laplace transform of the given function. 30. y(t) = t sin 3t 32. y(t) = t² cos 2t 27. y(t) = e²t cos 2t 29. y(t) = e-t (t² + 3t+4) In Exercises 30-33, use Proposition 2.14 to find the Laplace transform of the given function. 31. y(t) = te-t 33. y(t) = t²e²t In Exercises 34-41, use the propositions in Section 2 to trans- form the given initial value problem into an algebraic equation transform of y. involving L(y). Solve the resulting equation for the Laplace 34. y' + 2y = t sint, y(0) = 1 35. y' - y = 1²e-², y (0) = 0 36. y' + y = e' sin 3t, y(0) = 0 37. y - 2y = e²¹ cost, y(0) = -2 38. y" + 4y = f² sin 4r, y (0) = 0, y' (0) = -1
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