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5.6.1: Variation of Parameters (Exercises)

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    103512

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    In Exercises 1-6 find the general solution.

    1. \(y''+9y=\tan 3x\)

    2. \(y''+4y=\sin 2x\sec^2 2x\)

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

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

    5. \(y''-2y'+y=14x^{3/2}e^x\)

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

    In Exercises 7-29 find the general solution given the solutions \(y_{1}, y_{2}\) of the homogeneous equation.

    7. \(x^2y''+xy'- y=2x^2+2; \quad y_1=x, \quad y_2={1\over x}\)

    8. \({xy''+(2-2x)y'+(x-2)y=e^{2x}; \quad y_1=e^x, \quad y_2={e^x\over x}}\)

    9. \(4x^2y''+(4x-8x^2)y'+(4x^2-4x-1)y=4x^{1/2}e^x, \quad x > 0\); \(y_1=x^{1/2} e^x,\; y_2=x^{-1/2}e^x\)

    10. \(y''+4xy'+(4x^2+2)y=4e^{-x(x+2)};\quad y_1=e^{-x^2}, \quad y_2=xe^{-x^2}\)

    11. \(x^2y''-4xy'+6y=x^{5/2},\, x > 0;\quad y_1=x^2,\; y_2=x^3\)

    12. \(x^2y''-3xy'+3y=2x^4\sin x; \quad y_1=x,\; y_2=x^3\)

    13. \((2x+1)y''-2y'-(2x+3)y=(2x+1)^2e^{-x}; \quad y_1=e^{-x}, \quad y_2=xe^x\)

    14. \(4xy''+2y'+y=\sin\sqrt x; \quad y_1=\cos\sqrt x, \quad y_2=\sin\sqrt x\)

    15. \(xy''-(2x+2)y'+(x+2)y=6x^3e^x;\quad y_1=e^x,\quad y_2=x^3e^x\)

    16. \(x^2y''-(2a-1)xy'+a^2y=x^{a+1}; \quad y_1=x^a, \quad y_2=x^a \ln x\)

    17. \(x^2y''-2xy'+(x^2+2)y=x^3\cos x; \quad y_1=x\cos x, \quad y_2=x\sin x\)

    18. \(xy''-y'-4x^3y=8x^5;\quad y_1=e^{x^2},\; y_2=e^{-x^2}\)

    19. \((\sin x)y''+(2\sin x-\cos x)y'+(\sin x-\cos x)y=e^{-x}; \quad y_1=e^{-x},\quad y_2=e^{-x}\cos x\)

    20. \(4x^2y''-4xy'+(3-16x^2)y=8x^{5/2}; \quad y_1=\sqrt xe^{2x},\; y_2=\sqrt xe^{-2x}\)

    21. \(4x^2y''-4xy'+(4x^2+3)y=x^{7/2}; \quad y_1=\sqrt x\sin x,\; y_2=\sqrt x\cos x\)

    22. \(x^2y''-2xy'-(x^2-2)y=3x^4;\quad y_1=xe^x,\; y_2=xe^{-x}\)

    23. \(x^2y''-2x(x+1)y' +(x^2+2x+2)y=x^3e^x; \quad y_1=xe^x, \quad y_2=x^2e^x\)

    24. \(x^2y''-xy'-3y=x^{3/2}; \quad y_1=1/x, \quad y_2=x^3\)

    25. \(x^2y''-x(x+4)y'+2(x+3)y=x^4e^x; \quad y_1=x^2, \quad y_2=x^2e^x\)

    26. \(x^2y''-2x(x+2)y'+(x^2+4x+6)y=2xe^x; \quad y_1=x^2e^x, \quad y_2=x^3e^x\)

    27. \(x^2y''-4xy'+(x^2+6)y=x^4; \quad y_1=x^2\cos x, \quad y_2=x^2\sin x\)

    28. \((x-1)y''-xy'+y=2(x-1)^2e^x; \quad y_1=x, \quad y_2=e^x\)

    29. \(4x^2y''-4x(x+1)y'+(2x+3)y=x^{5/2}e^x; \quad y_1=\sqrt x, \quad y_2=\sqrt xe^x\)

    In Exercises 30-35 solve the initial value problem, given \(y_{1}\), \(y_{2}\) are solutions of the homogeneous equation.

    30. \((3x-1)y''-(3x+2)y'-(6x-8)y=(3x-1)^2e^{2x}, \quad y(0)=1,\; y'(0)=2\); \(y_1=e^{2x},\; y_2=xe^{-x}\)

    31. \((x-1)^2y''-2(x-1)y'+2y=(x-1)^2, \quad y(0)=3,\quad y'(0)=-6\); \(y_1=x-1\), \(y_2=x^2-1\)

    32. \((x-1)^2y''-(x^2-1)y'+(x+1)y=(x-1)^3e^x, \quad y(0)=4,\quad y'(0)=-6\); \(y_1=(x-1)e^x,\quad y_2=x-1\)

    33. \({(x^2-1)y''+4xy'+2y=2x, \quad y(0)=0,\; y'(0) =-2; \quad y_1={1\over x-1},\; y_2={1\over x+1}}\)

    34. \({x^2y''+2xy'-2y=-2x^2, \quad y(1)=1,\; y'(1)= -1; \quad y_1=x,\; y_2={1\over x^2}}\)

    35. \((x+1)(2x+3)y''+2(x+2)y'-2y=(2x+3)^2, \quad y(0)=0,\quad y'(0)=0\); \(y_1=x+2,\quad y_2={1\over x+1}\)

    36. Solve \(3y''-6y'+30y=15\sin x+e^x\tan 3x\)

    37. Solve \(y''+y=\sin x+\sec x\)

    38. Solve \(y''+3y'+2y=6x^2+{1\over 1+e^x}+\sin e^x\)

    39. Solve \(y''-2y'+y=x^3+4x+{e^x\over 1+x^2}\)

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

    This page titled 5.6.1: Variation of Parameters (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.