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I've been interested in a problem that involves a fairly simple looking set of differential equations:

$$ \frac{dx}{dt}=yz\;\;\;\frac{dy}{dt}=xz\;\;\;\frac{dz}{dt}=xy $$

From this it follows that $\frac{dx}{dy}=\frac{y}{x}$, $\frac{dy}{dz}=\frac{z}{y}$, etc.

Besides the trivial solutions of $x=0, y=0, z=0$, and where one of $x, y$ or $z$ equals some constant $k$ and the rest equal 0, I'm not sure how to continue with this.

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$\int x\text dx=\int y\text dy$ and therefore $y^2=x^2+b$ and, similarly, $z^2=x^2+c$, for constants $b$ and $c$.

That means that $$\int \frac{1}{\sqrt{x^2+b}\sqrt{x^2+c}}\text dx=\int \text dt.$$

You may be surprised to learn that this integral does not have an elementary solution.

You could however find various solutions to your system of differential equations by looking at special cases such as $b=c,b=0, ...$

For example with $b=c=0$ we have $$\int \frac{1}{x^2}\text dx=\int \text dt.$$ Then one solution is $x=y=z=-\frac{1}{t+a}$ and you might like to check this works.

Hope you can find some other interesting solutions.

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