Problems Find a general solution in terms of exponentials for each 31. u"+2u'+5u = 0 differential equation. 1. u" u'-6u 0 2. u"-9u=0 3. u" +9u=0 4. 4u" + 0 5. u"-4u'+4u = 0 6. u"+4u'+4u = 0 7. u"-4u4u = 0 8. u"+4u4u = 0 9. u" - 4u = 0 1 10. u" 4u'+8u = 0 11. u"+2u10u = 0 12. 2u"+6u+5u = 0 Write a general solution, in the form of Eq. 1.6.9 or 1.6.13, for each equation. 13. u" u'-6u= 0 14. u"-9u0 15. "+9u0 16. 4u" +u=0 17. u"-4u4u = 0 32. u" +5u3u = 0 Solve each initial-value problem. Express each answer in the form of Eq. 1.6.10, 1.6.13, or 1.6.19. 33. u" +9u0, u(0) = 0. u'(0) = 1 34. u" +5u'+6u = 0, u(0) = 2, u'(0) = 0 35. u"+4u'+4u = 0, u(0) = 0, u'(0) = 2 36. u"-4u 0, u(0) = 2, u'(0) = 1 Find the answer to each initial-value problem. Express each answer in the form of Eq. 1.6.9 or 1.6.13. 37. u"+9u0. = u (0) = 0, u'(0) = 1 38. u" +5u'+6u = 0, u(0) = 2, u'(0) = 0 39. u"-4u = 0, u (0) = = 2, u'(0) = 1 Determine the solution to each initial-value problem. Express each answer in the form Eq. 1.6.10 or 1.6.13. 40. u" + 9 = 0, u (0) = 0. u'(0) = 1 41. u"+5u6u = 0, u(0) = 2. u'(0) = 0 42. u"-4u0, u(0) = 2, u'(0) = 1 43. Consider the differential equation (n) (n-1) " +aill ++ An-14' + all=0