Please answer 10, 16, 19, 22, 37, 38 please!!
5.3 Linear Nonhomogeneous Equations 443 SECTION 5.3 EXERCISES Review Questions 1. Explain how to find the general solution of the nonhomogeneous 24. y' + 4y = e 'cost equation y" + py' + qy - f(r), where p and q are constants. 25. y" + 16y = / cos 2 2. What trial solution would you use to find a particular solution of the equation y" + py' + qy = r - 1 + 1. where p and q are 26. y" - 9y = (12 + 1)e- constants? 27. y"- y' - 2y = (cost - sin !) 3. What trial solution would you use to find a particular solution of 28. y" + 2y' - 4y = 12 cost the equation y" - 8y - 2 7 29-34. Undetermined coefficients with special cases Find a particu- What trial solution would you use to find a particular solution of lar solution of the following equations. the equation y" - 2y = 2 sin 3/7 29. = y = 3e' 5. What trial solution would you use to find a particular solution of the equation y" - 2y - be 'cos 41? 30. y+ y = 3 cost What trial solution would you use to find a particular solution of + y' - 6y = 4e- the equation y" - 2y = 1 + r + cos 317 32. y" + 4y = cos 21 7. What trial solution would you use to find a particular solution of 33. y' + 5y' + 6y - 2e-2 the equation y" - 4y = re 7 34. y" + 2y' + y = de-' 8. What trial solution would you use to find a particular solution of the equation y" - 4y' + 4y - 6e?? Further Explorations 35. Explain why or why not Determine whether the following state- Basic Skills ments are true and give an explanation or counterexample. 9-12. Undetermined coefficients with polynomials Find a particular solution of the following equations. a. To find a particular solution of the equation y" - 4y = 13, you should use the trial solution y - Ar. 9. - 9) = 27 + 1 b. To find a particular solution of the equation 10. y" - 2y' = 15y = 313 - 21 - 4 y" + y' - by = sin /, you should use the trial solution Yp - A sin t. 11. y' + 3y' - 18y = 274-12 c. To find a particular solution of the equation 12. y" + 5y = 613- 12 y" + 10y' + 25y = e", you should use the trial solution yo = Aest 13-16. Undetermined coefficients with exponentials Find a 36-41. Initial value problems Find the general solution of the follow- particular solution of the following equations. ing equations and then solve the given initial value problem. 13. y' - >= e 36. y - 9y = 2e . y(0) - 0,y'(0) = 0 14. y" + 3y = 5e-3 37. y" + y = 4 sin 21, y(0) = 1,y'(0) = 0 15. y" - 4y' - 32y = 60-3 38. y" + 3y' + 2y = 2e .y(0) = 0,y'(0) = 1 16. y" + 2y' - 8y = e- -2e' 39. y' + 4y' + 5y - 12, }(0) = 1.y'(0) =-1 17-20. Undetermined coefficients with sines/cosines Find a y" - y = 2e 'sin t, y(0) = 4. y'(0) = 0 particular solution of the following equations, 17. y" - y = 3 sin 21 41. y" + 9y = 6 cos 31, y(0) - 0. y'(0) = 0 18. y" + 4y = 5 cos 3t 42-45. Longer calculations Find a particular solution of the following equations. 19. y" - y' - by = sint + 3 cost 42. y" + 2y' + 2y - 5e cost 20. y' - 3y' = 4y = 2 cos 21 - 3 sin 21 43. y" + y' - y = 3 - 38+ 21-28. Undetermined coefficients with combined functions Find a 44. y" + 2y' + Sy = Be 'cos 2 particular solution of the following equations. 21. y" - 4y = 2el- 45. y" - y = 25te 'sin 3t y = cos 21 + 13 46-49. Higher-order equations The method of undetermined coeffi- 22. cients extends naturally to higher-order constant coefficient equations. 23. y" ty' - 12y = 21e ' Find a particular solution of the following equations. 47. "- 8y = Te
