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## System of coupled second order differential equations.

Hey folks I'm looking for a way to find the characteristic equation for a second order coupled system of differential equations such as...

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

$$\ddot{y} + C\dot{x} + Dy = 0$$

Where x and y are functions of time.

I know I can solve it by setting x and y to standard results (trig, exponential) but I'd like to know a method to solving this rather than plug and solve for coefficients.

Specifically I'd like to know how to find the characteristic equation for this. I've tried setting it to a first order system but I can't see it leading anywhere (or perhaps I just did it wrong...).

I don't want a full answer, just the name of a method or something like that.

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