# System of three non-linear differential equations

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.

$$\int x\text dx=\int y\text dy$$ 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 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.