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Thread: Variation of parameters question

  1. #1
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    Variation of parameters question

    Can someone help get me started with this I'm not sure where to begin..



    $\displaystyle
    \begin{gathered}
    x\frac{{dy}}
    {{dx}} + (2 + 3x)\frac{{dy}}
    {{dx}} + 3y = r(x); \hfill \\
    \hfill \\
    Solution \hfill \\
    y = 1/x \hfill \\
    with \hfill \\
    r(x) = 0 \hfill \\
    \end{gathered}
    $

    Form

    $\displaystyle
    y(x) = \frac{1}
    {x}v(x)
    $


    and I need to find the complete general equation corresponding to

    $\displaystyle
    r(x) = 3e^{ - 3x}
    $

    Can someone get me started with this.. do I differentiate y(x) part and substitute that into the equation?
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  2. #2
    MHF Contributor

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    Quote Originally Posted by dankelly07 View Post
    Can someone help get me started with this I'm not sure where to begin..



    $\displaystyle
    \begin{gathered}
    x\frac{{dy}}
    {{dx}} + (2 + 3x)\frac{{dy}}
    {{dx}} + 3y = r(x); \hfill \\$
    You mean $\displaystyle \frac{d^2y}{dx^2}+ (2+3x)\frac{dy}{dx}+ 3y= r(x)$

    $\displaystyle \hfill \\
    Solution \hfill \\
    y = 1/x \hfill \\
    with \hfill \\
    r(x) = 0 \hfill \\
    \end{gathered}
    $

    Form

    $\displaystyle
    y(x) = \frac{1}
    {x}v(x)
    $


    and I need to find the complete general equation corresponding to

    $\displaystyle
    r(x) = 3e^{ - 3x}
    $

    Can someone get me started with this.. do I differentiate y(x) part and substitute that into the equation?
    Because this is a second order equation, you will need two independent solutions to the homogeneous differential equation. Try putting y= v(x)/x into the differential equation. That reduces to a first order, separable equation so that you can find v and then another y solution.

    After you you have done that, you can use those solution in variation of parameters.
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