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Math Help - 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..



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

    Form

    <br />
y(x) = \frac{1}<br />
{x}v(x)<br />


    and I need to find the complete general equation corresponding to

    <br />
r(x) = 3e^{ - 3x} <br />

    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
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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..



    <br />
\begin{gathered}<br />
  x\frac{{dy}}<br />
{{dx}} + (2 + 3x)\frac{{dy}}<br />
{{dx}} + 3y = r(x); \hfill \\
    You mean \frac{d^2y}{dx^2}+ (2+3x)\frac{dy}{dx}+ 3y= r(x)

       \hfill \\<br />
  Solution \hfill \\<br />
  y = 1/x \hfill \\<br />
  with \hfill \\<br />
  r(x) = 0 \hfill \\ <br />
\end{gathered} <br />

    Form

    <br />
y(x) = \frac{1}<br />
{x}v(x)<br />


    and I need to find the complete general equation corresponding to

    <br />
r(x) = 3e^{ - 3x} <br />

    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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