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Math Help - Solving a DE by method of variation of parameters

  1. #1
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    Solving a DE by method of variation of parameters

    I have found the general solution for the equation of y'' - 9y = (x+1) e^{-x} to be

    y(x) = A e^{3x} + B e^{-3x} - \frac{1}{8} e^{-x} x - \frac{3 e^{-x}}{32}

    by the method of undetermined coefficients.

    However I need to use the method of variation of parameters, but I am totally confused.

    I have the homogeneous equation as: y_{h} = A e^{3x} + B e^{-3x} and so y_{p} = A(x) e^{3x} + B(x) e^{-3x}
    and
    A'(x) e^{3x} + B'(x) e^{-3x} = 0
    3A'(x) e^{3x} - 3B'(x) e^{-3x} = (x+1) e^{-x}

    But where do I go now? Can anyone help me with the steps to solve this?

    Thank you all in advance.
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Solving a DE by method of variation of parameters

    Try multiplying the first equation by 3, then adding the two equations, as this will eliminate B'(x)e^{-3x}.
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