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Math Help - Method of undetermined coefficients - Nonhomogeneous Linear Systems

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    Junior Member utopiaNow's Avatar
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    Method of undetermined coefficients - Nonhomogeneous Linear Systems

    Hello,

    I am simply trying to understand the way this system was solved in the text:

    <br />
x^{\prime} = \left(\begin{array}{cc}2&-1\\3&-2\end{array}\right)x\ +\ \left(\begin{array}{c}e^t\\t\end{array}\right)<br />


    The two eigenvalues and corresponding eigenvectors that were found were:

    <br />
\lambda_{1,2} = 1, -1<br />

    <br />
\xi^{(1)} = \left(\begin{array}{c}1\\1\end{array}\right)\ \xi^{(2)} = \left(\begin{array}{c}1\\3\end{array}\right)<br />

    Therefore we know the homogeneous part of the solution. Now Since:
    <br />
g(t) = \left(\begin{array}{c}1\\0\end{array}\right)e^t + \left(\begin{array}{c}0\\1\end{array}\right)t<br />

    we guess by method of undetermined co-efficients that the particular solution has the form:

    <br />
x_p(t) = \mathbf ate^t + \mathbf be^t + \mathbf ct + \mathbf d<br />

    So now sub this into the original Equation which will give:
     \frac{dx_p}{dt} =  \mathbf Ax_p + \mathbf g(t)
    and collect terms to get:
    <br />
\mathbf {Aa} = \mathbf{a} <br />

    <br />
\mathbf {Ab} = \mathbf a + \mathbf b - \left(\begin{array}{c}1\\0\end{array}\right)<br />

    etc.

    I understand how to find a, c, d. But I'm not following the steps needed to find b, I know that we can find a because it is an eigenvalue of A, but with this info. how do we now find b?

    Thanks in Advance

    Last edited by utopiaNow; March 23rd 2009 at 06:00 PM.
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