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Math Help - solving a system by substitution 3b

  1. #1
    MHF Contributor
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    solving a system by substitution 3b

    solve this system by sustituting  t=e^u and building a function y(u)=x(e^u)
     <br />
t\frac{dx}{dt}=(\begin{array}{cc}<br />
2 & 2\\<br />
1 & 3\end{array})x<br />
    <br />
x(1)=(\begin{array}{c}<br />
2\\<br />
3\end{array})<br />

    how to do it?
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  2. #2
    Super Member PaulRS's Avatar
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    Apply the chain rule to get: \displaystyle \frac{dy}{du} = e^u \cdot \frac{dx}{dt} = t \cdot \frac{dx}{dt} ( \frac{dx}{dt} evaluated at e^u = t )

    Then you have \dot{y} = \begin{pmatrix}<br />
2 & 2\\ <br />
1 & 3<br />
\end{pmatrix} \cdot y now you could solve it, say, by computing the matrix exponential.

    For the initial values think what knowing x(1) would mean in terms of y, at the end, when you have y you can undo the change of variables to get the solution for t>0.
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  3. #3
    MHF Contributor
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    you sat that y=x(u)t(u)=x(u)e^y
    so the derivative is must have sum of two member
    why you have one?
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