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Solving second order coupled differential equation

  1. Mar 27, 2014 #1
    How do we solve a system of coupled differential equations written below?
    [tex]
    -\frac{d^2}{dr^2}\left(
    \begin{array}{c}
    \phi_{l,bg}(r) \\
    \phi_{l,c}(r) \\
    \end{array} \right)+ \left(
    \begin{array}{cc}
    f(r) & \alpha_1 \\
    \alpha_2 & g(r)\\
    \end{array} \right).\left(
    \begin{array}{c}
    \phi_{l,bg}(r) \\
    \phi_{l,c}(r) \\
    \end{array} \right) = E\left(
    \begin{array}{c}
    \phi_{l,bg}(r) \\
    \phi_{l,c}(r) \\
    \end{array} \right)
    [/tex]

    Here [itex]f(r)[/itex] and [itex]g(r)[/itex] are quadratic functions of [itex]r[/itex]. [itex]\alpha_1,\alpha_2\text{ and }E[/itex] are constants.
     
  2. jcsd
  3. Mar 27, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    Your system can be reduced to the first order system
    [tex]
    y' = A(r)y
    [/tex]
    where
    [tex]
    y = \begin{pmatrix} \phi_{l,bg} \\ \phi_{l,c} \\ \phi_{l,bg}' \\ \phi_{l,c}' \end{pmatrix}
    [/tex]
    and
    [tex]
    A(r) = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ E - f(r) & -\alpha_1 & 0 & 0\\
    -\alpha_2 & E - g(r) & 0 & 0\end{pmatrix}
    [/tex]
    which has a solution in terms of a Magnus series.
     
  4. Mar 27, 2014 #3
    Thank you for the help.
     
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