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Math Help - non-homogeneous Cauchy-Euler problem

  1. #1
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    non-homogeneous Cauchy-Euler problem

    I think I'm doing this right, but I'm not getting the right answer it seems.

    Use variation of parameters to find a particular solution of the non-homogeneous Cauchy-Euler problem:

    x^2y''-2xy'+2y=x^4e^x

    I found the roots to be 1,2 so my general solution is y=c_{1}x+c_{2}x^2+y_{p}

    Then I used variation of parameters. I found W, the Wronskian of y_{1} and y_{2}, to be x^2 and W_{1} and W_{2} to be -x^{6}e^x and x^{5}e^x respectively.

    Dividing W_{1} and W_{2} by W I get u_{1}' as -x^{4}e^x and u_{2}' as x^{3}e^x

    Integrating I get u_{1}=-e^{x}(x^4-4x^3+12x^2-24x+24) and u_{2}=e^{x}(x^3-3x^2+6x-6)

    So my y_{p} should be -xe^{x}(x^4-4x^3+12x^2-24x+24)+x^{2}e^{x}(x^3-3x^2+6x-6)

    Whenever I put this in, it says it's not correct. If there's something I'm missing or that I did incorrectly, any help would be appreciated.
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  2. #2
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    For the method to work, the coefficient of the second derivative has to be unity. You have to start the calculation of the P.I. by dividing throughout by x^2.
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  3. #3
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    Thanks! It worked now.
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