I have to derive equations of motion from Lagrangian and stumbled upon the following system of equations (constants are simplified, that information is unneeded)(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

\begin{cases}

\ddot{x}-A\dot{y}+Bx=0 \\

\ddot{y}+A\dot{x}+Dy=0

\end{cases}

[/itex]

This is an extension of a simpler problem where B=D. There I just multiplied the second equation by i, added equations together, substituted z=x+iy and solved for z.

But doing the same with current system doesn't help much (can't substitute z=x+iy, b/c [itex]B\ne{D}[/itex]).

When choosing E=(B+D)/2 and F=E-B then

[itex]

\begin{cases}

\ddot{x}-A\dot{y}+Ex=Fx\\

\ddot{y}+A\dot{x}+Ey=-Fy

\end{cases}

[/itex]

isn't much help either.

I know there are some techniques for solving coupled differential equations like writing a reciprocal matrix for the system but it seems that it applies only to 1st order ODEs.

Is there any analytic solutions to this?

When punching it into Maple, it throws a huge block of square roots, although simpler problem gave me a combination of exponentials (for z). I know this problem must give similar answer with little modifications, but don't know how to tackle it.

Help and pointers much appreciated

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# Solving system of 2nd order coupled ODEs

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