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## Homework Statement

I need to (analytically) solve a system of coupled second-order ODEs:

(A) [itex]\frac{du}{dt} - fv = \Omega^2x[/itex]

(B) [itex]\frac{dv}{dt} + fu = \Omega^2y[/itex]

where

[itex]u = \frac{dx}{dt}[/itex]

[itex]v = \frac{dy}{dt}[/itex]

subject to the initial conditions [itex]u(t=0) = U[/itex] and [itex]v(t=0) = 0[/itex].

## Homework Equations

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## The Attempt at a Solution

(1) I first converted the ODEs to:

[itex]\frac{d^2x}{dt^2} - f\frac{dy}{dt} = \Omega^2x[/itex]

[itex]\frac{d^2y}{dt^2} + f\frac{dx}{dt} = \Omega^2y[/itex]

and then added and subtracted them to get:

[itex]\frac{d^2(x+y)}{dt^2} - f\frac{d(x-y)}{dt} = \Omega^2(x+y)[/itex]

[itex]\frac{d^2(x-y)}{dt^2} - f\frac{d(x+y)}{dt} = \Omega^2(x-y)[/itex]

then making the substitution

[itex]\alpha = x + y[/itex]

[itex]\beta = x - y[/itex]

which just leads me to the exact problem I started with.

Since I got stuck here, I tried it a different way...

(2) Making note that, from (B), [itex]u=\frac{1}{f}(\Omega^2y - \frac{dv}{dt})[/itex]. Plugging into A, we get:

[itex]\frac{1}{f}(\Omega^2\frac{dy}{dt} - \frac{d^2v}{dt^2})-fv=\Omega^2x[/itex]

(...after rearranging...)

[itex]\frac{d^2v}{dt^2} + v(f-\frac{\Omega^2}{f}) = -\Omega^2x[/itex]

Since [itex]v=\frac{dy}{dt}[/itex], this is now a

*third*order coupled ODE (after we do something similar to above for [itex]u[/itex]), and I don't know how to solve it.

Does anyone know where to go from here?