# 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.

Actually I think I've got it...

Setting $$x_1 = x$$, $$y_1 = y$$, then converting into four first order differential equations. Find eigenvalues of 4x4 matrix etc... Think that's it.

vela
Staff Emeritus
Homework Helper
It's not entirely clear what you did other than add a subscript to x and y, but I suspect you have the right approach.

lurflurf
Homework Helper
rewrite the system (I will use E for your D and D for differentiation)