# Initial condition leads to periodic solution

## Main Question or Discussion Point

Hi, I have a question that asks me to explain how you could know if the initial conditions lead to a periodic solution of an ODE but I have no clue right now. I'd appreciate any help.

Thanks

edit: The question is a bit obscure because I am not sure if they are asking in general of for the specific equations that were given so I decided to copy the exact question in here.

x'' = 2*y' + x - (1 - (1/82.5))*(x+(1/82.5))/sqrt((x+mu)^2 + y^2)^3 - (1/82.5)*(x-(1 - (1/82.5)))/sqrt((x-muBar)^2 + y^2)^3 - f*x'

y'' = -2*x' + y - (1 - (1/82.5))*y/sqrt((x+mu)^2 + y^2)^3 - (1/82.5)*y/sqrt((x-muBar)^2 + y^2)^3 - f*y';

where f is any constant initialy 0

and the question is

Discuss how you could determine whether a given set of initial condtions leads to a periodic solution

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Well, i have never seen these kind of diff. eq. that include the derivatives of bot x and y.
Is this a partial derivative, or an ordinary diff. eq?
This is out of my domain...lol...

But i think that the question is specific for this equation, not in general.

For a local solution, linearize around the equilibrium and check the eigenvalues if they are on the imaginary axis...

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