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Math Help - The Method of Elimination

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    The Method of Elimination

    FIND THE GENERAL SOLUTION OF THE LINEAR SYSTEM

    x'' = 6x + 2y, y'' = 3x + 7y
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    Quote Originally Posted by bearej50 View Post
    FIND THE GENERAL SOLUTION OF THE LINEAR SYSTEM

    x'' = 6x + 2y, y'' = 3x + 7y
    Define u= x' and v= y'. Then the differential equation x"= 6x+ 2y becomes u'= 6x+ 2y and y"= 3x+ 7y becomes v'= 3x+ 7y.

    So instead of two second order equations we have four first order equations:
    x'= u, y'= v, u'= 6x+ 2y, and v'= 3x+ 7y which we could also write as the single first order matrix equation:
    \frac{d\begin{bmatrix}x \\ y \\ u \\ v\end{bmatrix}}{dt}= \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0& 1 \\ 6 & 2 & 0 & 0 \\3 & 7 & 0 & 0\end{bmatrix}\begin{bmatrix}x \\ y \\ u \\ v\end{bmatrix}

    And the first step in that is to find the eigenvalues and eigenvectors of the coefficient matrix.
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    Quote Originally Posted by bearej50 View Post
    FIND THE GENERAL SOLUTION OF THE LINEAR SYSTEM





    x'' = 6x + 2y, y'' = 3x + 7y
    As the title of the post suggests, if we let y = \frac{x''-6x}{2} (**) and eliminate y from the second ODE we obtain x^{(4)} - 12x'' + 36 = 0. Its characteristic equation is m^4 - 13m^2 + 36 = 0 or (m^2-4)(m^2-9)=0 which has solution m = \pm2, \pm3 and so the solution is

     <br />
x = c_1 e^{-2x} + c_2 e^{2x} + c_3 e^{-3x} + c_4 e^{3x}<br />

    Once you have this, sub into your y above (**).
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