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Given the following equations

\begin{cases} \dot x = - x - y + \cos t,\\ \dot y = x - y + \sin t. \end{cases}

I am looking for periodic orbits. I managed to find the general solution for some initial conditions $x_0,y_0$ \begin{equation} \begin{pmatrix} x \\ y \end{pmatrix} = e^{-t} \begin{pmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{pmatrix} \begin{pmatrix} x_0-1 \\ y_0 \end{pmatrix} + \begin{pmatrix} \cos t \\ \sin t \end{pmatrix} \end{equation}

However, I am not sure what to do and how to proceed to finding the periodic orbits. One way of doing that seems to be choosing the initial conditions so that the solution is a periodic function, so $x_0 = 1, y_0=0$ but it seems odd... Can someone suggest any hints/steps?

Thank you.

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You got it exactly right. Any terms with $e^{-t} \cos(t)$ or $e^{-t} \sin(t)$ will spoil the periodicity, so you do in fact need $x_0 = 1$ and $y_0 = 0$ to have a periodic solution.

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