Solving second order coupled differential equation

[Ravi Mohan]

How do we solve a system of coupled differential equations written below?
<br /> -\frac{d^2}{dr^2}\left(<br /> \begin{array}{c}<br /> \phi_{l,bg}(r) \\<br /> \phi_{l,c}(r) \\<br /> \end{array} \right)+ \left(<br /> \begin{array}{cc}<br /> f(r) &amp; \alpha_1 \\<br /> \alpha_2 &amp; g(r)\\ <br /> \end{array} \right).\left(<br /> \begin{array}{c}<br /> \phi_{l,bg}(r) \\<br /> \phi_{l,c}(r) \\<br /> \end{array} \right) = E\left(<br /> \begin{array}{c}<br /> \phi_{l,bg}(r) \\<br /> \phi_{l,c}(r) \\<br /> \end{array} \right) <br />

Here f(r) and g(r) are quadratic functions of r. \alpha_1,\alpha_2\text{ and }E are constants.

 

[pasmith]

How do we solve a system of coupled differential equations written below?
<br /> -\frac{d^2}{dr^2}\left(<br /> \begin{array}{c}<br /> \phi_{l,bg}(r) \\<br /> \phi_{l,c}(r) \\<br /> \end{array} \right)+ \left(<br /> \begin{array}{cc}<br /> f(r) &amp; \alpha_1 \\<br /> \alpha_2 &amp; g(r)\\ <br /> \end{array} \right).\left(<br /> \begin{array}{c}<br /> \phi_{l,bg}(r) \\<br /> \phi_{l,c}(r) \\<br /> \end{array} \right) = E\left(<br /> \begin{array}{c}<br /> \phi_{l,bg}(r) \\<br /> \phi_{l,c}(r) \\<br /> \end{array} \right) <br />

Here f(r) and g(r) are quadratic functions of r. \alpha_1,\alpha_2\text{ and }E are constants.

Your system can be reduced to the first order system
<br /> y&#039; = A(r)y<br />
where
<br /> y = \begin{pmatrix} \phi_{l,bg} \\ \phi_{l,c} \\ \phi_{l,bg}&#039; \\ \phi_{l,c}&#039; \end{pmatrix}<br />
and
<br /> A(r) = \begin{pmatrix} 0 &amp; 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 1 \\ E - f(r) &amp; -\alpha_1 &amp; 0 &amp; 0\\<br /> -\alpha_2 &amp; E - g(r) &amp; 0 &amp; 0\end{pmatrix}<br />
which has a solution in terms of a Magnus series.

 

[Ravi Mohan]

Thank you for the help.