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Math Help - How to solve this non-homogenous second order ODE?

  1. #1
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    How to solve this non-homogenous second order ODE?

    y''-2y'+y=xe^xlnx

    I don't know what I should do because of the logarithm function, the exercise has come after introducing the undetermined coefficients methods, so I assume it should be solved that way but I don't know how.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: How to solve this non-homogenous second order ODE?

    Quote Originally Posted by Nikita2011 View Post
    y''-2y'+y=xe^xlnx

    I don't know what I should do because of the logarithm function, the exercise has come after introducing the undetermined coefficients methods, so I assume it should be solved that way but I don't know how.
    The general solution has the form y(x)=C_1(x)e^x+C_2(x)xe^x . Have you covered the method of variation of parameters?
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    Re: How to solve this non-homogenous second order ODE?

    Quote Originally Posted by FernandoRevilla View Post
    The general solution has the form y(x)=C_1(x)e^x+C_2(x)xe^x . Have you covered the method of variation of parameters?
    well, that's obviously the general solutions of the homogenous question, but how to find y_p.

    I don't know what the method of variation of parameters is, but I solved it through the method that we take y_p = v_1y_1 + v_2y_2 and then we look for v_1,v_2. It was finally solved but I wonder why the author has put it in the section of undetermined coefficients method, is it possible to solve this with that method too? I doubt.
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  4. #4
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    Re: How to solve this non-homogenous second order ODE?

    It is possible to solve via the method of undetermined coefficients by trying a y_p of the form

    y = ax^3 e^x \ln x + b x^3 e^x.
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