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Consider the following system of differential equations:

$\frac{dx}{dt}=y-1$

$\frac{dy}{dt}=-xy$

So I figured that, if the initial conditions satisfy $(x_0,y_0)\ne(0,1)$, I can rewrite the system as

$\frac{dy}{dx}=\frac{-xy}{y-1}$

and for $y\ne0$ this can be rewritten as

$\frac{dy}{dx}=\frac{-x}{\big(\frac{y-1}{y}\big)}$,

thus using separation of variables yields the expression

$y-\ln|y|-y_0+\ln|y_0|=-\frac12x^2+\frac12(x_0)^2$.

But I don't see how this expression can give me a solution to the aforementioned system, since I believe $y$ cannot be written explicitley as a function of $x$. I guess I could express $x$ as a function of $y$, but would this give me a valid solution as well? Also, I'm not sure how to determine periodicity. I know from a plot that there are periodic solutions (for $y_0>0$, except in the stationary point $(0,1)$), but I don't understand how I can find them if I don't have an explicit solution to the system. Or how I can find them at all, actually, if I would have a function $y$ in terms of $x$. I have found this page dealing with the same problem, asking roughly the same question. However, I do not seem to understand the argument given in the answer. It's not something that has yet been discussed in my course, too. Any help would be appreciated!

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