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Mathematics LibreTexts

Section 5.3 Answers

  • Page ID
    28908
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    1. \(y_{p}=-1+2x+3x^{2};\: y=-1+2x+3x^{2}+c_{1}e^{-6x}+c_{2}e^{x}\)

    2. \(y_{p}=1+x;\: y=1+x+e^{2x}(c_{1}\cos x+c_{2}\sin x)\)

    3. \(y_{p}=-x+x^{3};\: y=-x+x^{3}+c_{1}e^{-7x}+c_{2}e^{-x}\)

    4. \(y_{p} = 1 − x^{2};\: y = 1 − x^{2} + e^{2x} (c_{1} + c_{2}x)\)

    5. \(y_{p} = 2x + x^{3};\: y = 2x + x^{3} + e^{−x} (c_{1} \cos 3x + c_{2} \sin 3x);\: y = 2x + x^{3} + e^{−x} (2 \cos 3x + 3 \sin 3x)\)

    6. \(y_{p} = 1 + 2x;\: y = 1 + 2x + e^{−3x} (c_{1} \cos x + c_{2} \sin x);\: y = 1 + 2x + e^{−3x} (\cos x − \sin x)\)

    8. \(y_{p}=\frac{2}{x}\)

    9. \(y_{p}=4x^{1/2}\)

    10. \(y_{p}=\frac{x^{3}}{2}\)

    11. \(y_{p}=\frac{1}{x^{3}}\)

    12. \(y_{p}=9x^{1/3}\)

    13. \(y_{p}=\frac{2x^{4}}{13}\)

    16. \(y_{p}=\frac{e^{3x}}{3};\: y=\frac{e^{3x}}{3}+c_{1}e^{-6x}+c_{2}e^{x}\)

    17. \(y_{p} = e^{2x};\: y = e^{2x} (1 + c_{1}\cos x + c_{2}\sin x)\)

    18. \(y = −2e^{−2x};\: y = −2e^{−2x} + c_{1}e^{−7x} + c_{2}e^{−x};\: y = −2e^{−2x} − e^{−7x} + e^{−x}\)

    19. \(y_{p} = e^{x};\: y = e^{x} + e^{2x} (c_{1} + c_{2}x);\: y = e^{x} + e^{2x} (1 − 3x)\)

    20. \(y_{p}=\frac{4}{25}e^{x/2};\: y=\frac{4}{45}e^{x/2}+e^{-x}(c_{1}\cos 3x+c_{2}\sin 3x)\)

    21. \(y_{p} = e^{−3x};\: y = e^{−3x} (1 + c_{1}\cos x + c_{2}\sin x)\)

    24. \(y_{p} = \cos x − \sin x;\: y = \cos x − \sin x + e^{4x} (c_{1} + c_{2}x)\)

    25. \(y_{p} = \cos 2x − 2 \sin 2x;\: y = \cos 2x − 2 \sin 2x + c_{1} + c_{2}e^{−x}\)

    26. \(y_{p}=\cos 3x;\: y=\cos 3x+e^{x}(c_{1}\cos\sqrt{2}x+c_{2}\sin\sqrt{2}x)\)

    27. \(y_{p} = \cos x + \sin x;\: y = \cos x + \sin x + e^{−3x} (c_{1} \cos 2x + c_{2} \sin 2x)\)

    28. \(y_{p} = −2 \cos 2x + \sin 2x;\: y = −2 \cos 2x + \sin 2x + c_{1}e^{−4x} + c_{2}e^{−3x};\: y = −2 \cos 2x + \sin 2x + 2e^{−4x} − 3e^{−3x}\)

    29. \(y_{p} = \cos 3x − \sin 3x;\: y = \cos 3x − \sin 3x + e^{3x} (c_{1} + c_{2}x)\: y = \cos 3x − \sin 3x + e^{3x} (1 + 2x)\)

    30. \(y=\frac{1}{\omega _{0}^{2}-\omega ^{2}}(M\cos\omega x+N\sin\omega x)+c_{1}\cos\omega_{0}x+c_{2}\sin\omega_{0}x\)

    33. \(y_{p}=-1+2x+3x^{2}+\frac{e^{3x}}{3};\: y=-1+2x+3x^{2}+\frac{e^{3x}}{3}+c_{1}e^{-6x}+c_{2}e^{x}\)

    34. \(y_{p} = 1 + x + e^{2x};\: y = 1 + x + e^{2x} (1 + c_{1}\cos x + c_{2}\sin x)\)

    35. \(y_{p} = −x + x^{3} − 2e^{−2x};\: y = −x + x^{3} − 2e^{−2x} + c_{1}e^{−7x} + c_{2}e^{−x}\)

    36. \(y_{p} = 1 − x^{2} + e^{x};\: y = 1 − x^{2} + e^{x} + e^{2x} (c_{1} + c_{2}x)\)

    37. \(y_{p}=2x+x^{3}+\frac{4}{45}e^{x/2};\: y=2x+x^{3}+\frac{4}{45}e^{x/2}+e^{-x}(c_{1}\cos 3x+c_{2}\sin 3x)\)

    38. \(y_{p} = 1 − x^{2} + e^{x};\: y = 1 − x^{2} + e^{x} + e^{2x} (1+c_{1}\cos x + c_{2}\sin x)\)