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Math Help - Solve using method of variation of parameters

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    Solve using method of variation of parameters

    Maybe you can help me with this ODE question? It says to use the method of variation of parameters to find the general solution to

    y'' + xy' + y = xe^(-x^2/2) given that one solution is e^(-x^2/2) and the other is that sum of series.

    The only way I know how to do this is by solving for two functions v1' and v2' but the series is throwing me off.

    If you have any ideas that would be great.
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  2. #2
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    Quote Originally Posted by sarah232 View Post
    Maybe you can help me with this ODE question? It says to use the method of variation of parameters to find the general solution to

    y'' + xy' + y = xe^(-x^2/2) given that one solution is e^(-x^2/2) and the other is that sum of series.

    The only way I know how to do this is by solving for two functions v1' and v2' but the series is throwing me off.

    If you have any ideas that would be great.
    Using a variation of parameters could get a little cumbersome, especially with one solution a series solution. Another way is to use a reduction of order type idea. Let

    y = u \,e^{-x^2/2}

    which substituting into your ODE gives

     <br />
u'' - xu' -x = 0.<br />

    After two integrations you should see your second solution emerge in addition to your particular solution.
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