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Math Help - Linear System: Phase Planes

  1. #1
    Junior Member utopiaNow's Avatar
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    Linear System: Phase Planes

    Hi Everyone,

    The system given is:

    <br /> <br />
\frac{dX}{dt} = \left(\begin{array}{cc}5&-1\\3&1\end{array}\right)X<br /> <br /> <br />

    The eigenvalues and the corresponding eigenvectors I found were:

    <br />
r_1 = 2, r_2 = 4<br /> <br />

    <br />
\xi_1 = (1, 3)^T, \xi_2 = (1, 1)^T<br />

    Therefore,

    <br />
x = c_1\left(\begin{array}{c}1\\3\end{array}\right)e^{  2t} + c_2\left(\begin{array}{c}1\\2\end{array}\right)e^{  4t}<br />


    Now the text says to sketch several trajectories in the phase plane and also sketch some typical graphs of x_1 versus t .

    The text does not really explain how its generating the example graphs, it just shows pictures of general graphs. How would I go about sketching these? Without using a computer of course.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by utopiaNow View Post
    Hi Everyone,

    The system given is:

    <br /> <br />
\frac{dX}{dt} = \left(\begin{array}{cc}5&-1\\3&1\end{array}\right)X<br /> <br /> <br />

    The eigenvalues and the corresponding eigenvectors I found were:

    <br />
r_1 = 2, r_2 = 4<br /> <br />

    <br />
\xi_1 = (1, 3)^T, \xi_2 = (1, 1)^T<br />

    Therefore,

    <br />
x = c_1\left(\begin{array}{c}1\\3\end{array}\right)e^{  2t} + c_2\left(\begin{array}{c}1\\2\end{array}\right)e^{  4t}<br />


    Now the text says to sketch several trajectories in the phase plane and also sketch some typical graphs of x_1 versus t .

    The text does not really explain how its generating the example graphs, it just shows pictures of general graphs. How would I go about sketching these? Without using a computer of course.
    First graph the two lines given by the eigenvectors: <1, 3> lies along the line y= 3x and <1, 1> along the line y= x. Since both eigenvalues are positive, the "flow" is outward. All other solutions will be outward along curves emanating from the origin curving toward y= x because it corresponds to the larger eigenvalue, 4 so the flow is "faster" there.
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