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5.5.1: Annihilation (Exercises)

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    In Exercises 1-51 find the general solution.

    1. \(y''-9y=54\)

    2. \(y''+y'=3\)

    3. \(y''+4y'+4y=2x+6\)

    4. \(y''-y'-12y=e^{4x}\)

    5. \(y''-2y'-3y=4e^x-9\)

    6. \(y''+25y=6\sin x\)

    7. \(y''+6y'+9y=-xe^{4x}\)

    8. \(y''-y=x^2e^x+5\)

    9. \(y''-2y'+5y=e^x\sin x\)

    10. \(y''+25y=20\sin 5x\)

    11. \(y''+y'+y=x\sin x\)

    12. \(y''-3y'+2y=e^{3x}(1+x)\)

    13. \(y''-6y'+5y=e^{-3x}(35-8x)\)

    14. \(y''-2y'-3y=e^x(-8+3x)\)

    15. \(y''+2y'+y=e^{2x}(-7-15x+9x^2)\)

    16. \(y''+4y=e^{-x}(7-4x+5x^2)\)

    17. \(y''-y'-2y=e^x(9+2x-4x^2)\)

    18. \(y''-4y'-5y=-6xe^{-x}\)

    19. \(y''-3y'+2y=e^x(3-4x)\)

    20. \(y''+y'-12y=e^{3x}(-6+7x)\)

    21. \(2y''-3y'-2y=e^{2x}(-6+10x)\)

    22. \(y''+2y'+y=e^{-x}(2+3x)\)

    23. \(y''-2y'+y=e^x(1-6x)\)

    24. \(y''-4y'+4y=e^{2x}(1-3x+6x^2)\)

    25. \(9y''+6y'+y=e^{-x/3}(2-4x+4x^2)\)

    26. \(y''-3y'+2y=e^{3x}(1+x)\)

    27. \(y''-6y'+8y=e^x(11-6x)\)

    28. \(y''+6y'+9y=e^{2x}(3-5x)\)

    29. \(y''+2y'-3y=-16xe^x\)

    30. \(y''-2y'+y=e^x(2-12x)\)

    31. \(y''+3y'+2y=7\cos x-\sin x\)

    32. \(y''+3y'+y=(2-6x)\cos x-9\sin x\)

    33. \(y''+2y'+y=e^x(6\cos x+17\sin x)\)

    34. \(y''+3y'-2y=-e^{2x}(5\cos2x+9\sin2x)\)

    35. \(y''-y'+y=e^x(2+x)\sin x\)

    36. \(y''+3y'-2y=e^{-2x}\left[(4+20x)\cos 3x+(26-32x)\sin 3x\right]\)

    37. \(y''+4y=-12\cos2x-4\sin2x\)

    38. \(y''+y=(-4+8x)\cos x+(8-4x)\sin x\)

    39. \(4y''+y=-4\cos x/2-8x\sin x/2\)

    40. \(y''+2y'+2y=e^{-x}(8\cos x-6\sin x)\)

    41. \(y''-2y'+5y=e^x\left[(6+8x)\cos 2x+(6-8x)\sin2x\right]\)

    42. \(y''+2y'+y=8x^2\cos x-4x\sin x\)

    43. \(y''+3y'+2y=(12+20x+10x^2)\cos x+8x\sin x\)

    44. \(y''+3y'+2y=(1-x-4x^2)\cos2x-(1+7x+2x^2)\sin2x\)

    45. \(y''-5y'+6y=-e^x\left[(4+6x-x^2)\cos x-(2-4x+3x^2)\sin x\right]\)

    46. \(y''-2y'+y=-e^x\left[(3+4x-x^2)\cos x+(3-4x-x^2)\sin x\right]\)

    47. \(y''-2y'+2y=e^x\left[(2-2x-6x^2)\cos x+(2-10x+6x^2)\sin x\right]\)

    48. \(y''+2y'+y=e^{-x}\left[(5-2x)\cos x-(3+3x)\sin x\right]\)

    49. \(y''+9y=-6\cos 3x-12\sin 3x\)

    50. \(y''+3y'+2y=(1-x-4x^2)\cos2x-(1+7x+2x^2)\sin2x\)

    51. \(y''+4y'+3y=e^{-x}\left[(2+x+x^2)\cos x+(5+4x+2x^2)\sin x\right]\)

    In Exercises 52-66 solve the initial value problem.

    52. \(y''-64y=16, \quad y(0)=1,\quad y'(0)=0\)

    53. \(y''-5y'=x-2, \quad y(0)=0,\quad y'(0)=2\)

    54. \(y''+y=8\cos 2x-4\sin x, \quad y({\pi\over 2})=-1,\quad y'({\pi\over 2})=0\)

    55. \(y''-4y'+8y=x^3, \quad y(0)=2,\quad y'(0)=4\)

    56. \(y''-4y'-5y=9e^{2x}(1+x), \quad y(0)=0,\quad y'(0)=-10\)

    57. \(y''+3y'-4y=e^{2x}(7+6x), \quad y(0)=2,\quad y'(0)=8\)

    58. \(y''+4y'+3y=-e^{-x}(2+8x), \quad y(0)=1,\quad y'(0)=2\)

    59. \(y''-3y'-10y=7e^{-2x}, \quad y(0)=1,\quad y'(0)=-17\)

    60. \(y''-7y'+6y=-e^x(17\cos x-7\sin x), \quad y(0)=4,\; y'(0)=2\)

    61. \(y''-2y'+2y=-e^x(6\cos x+4\sin x), \quad y(0)=1,\; y'(0)=4\)

    62. \(y''+6y'+10y=-40e^x\sin x, \quad y(0)=2,\quad y'(0)=-3\)

    63. \(y''-6y'+10y=-e^{3x}(6\cos x+4\sin x), \quad y(0)=2,\quad y'(0)=7\)

    64. \(y''-3y'+2y=e^{3x}\left[21\cos x-(11+10x)\sin x\right], \; y(0)=0, \quad y'(0)=6\)

    65. \(y''+4y=-e^{-2x}\left[(4-7x)\cos x+(2-4x)\sin x\right], \; y(0)=3, \quad y'(0)=1\)

    66. \(y''+4y'+4y=2\cos2x+3\sin2x+e^{-x}, \quad y(0)=-1,\; y'(0)=2\)

    In Exercises 67-72 solve a. and b. and use the principle of superposition to solve c.

    67. a. \(y''+y'+y=xe^x\)

    b. \(y''+y'+y=e^{-x}(1+2x)\)

    c. \(y''+y'+y=xe^x+e^{-x}(1+2x)\)

    68. a. \(y''-7y'+12y=-e^x(17-42x)\)

    b. \(y''-7y'+12y=-e^{3x}\)

    c. \(y''-7y'+12y=-e^x(17-42x)-e^{3x}\)

    69. a. \(y''-8y'+16y=6xe^{4x}\)

    b. \(y''-8y'+16y=2+16x+16x^2\)

    c. \(y''-8y'+16y=6xe^{4x}+2+16x+16x^2\)

    70. a. \(y''-3y'+2y=-e^{2x}(3+4x)\)

    b. \(y''-3y'+2y=-e^x\)

    c. \(y''-3y'+2y=-e^{2x}(3+4x)-e^x\)

    71. a. \(y''-2y'+2y=e^x(1+x)\)

    b. \(y''-2y'+2y=e^{-x}(2-8x+5x^2)\)

    c. \(y''-2y'+2y=e^x(1+x)+e^{-x}(2-8x+5x^2)\)

    72. a. \(y''+y=e^{-x}(2-4x+2x^2)\)

    b. \(y''+y=e^{3x}(8-12x-10x^2)\)

    c. \(y''+y=e^{-x}(2-4x+2x^2)+e^{3x}(8-12x-10x^2)\)

    This page titled 5.5.1: Annihilation (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.