I would like to solve the following system of two coupled nonlinear ODEs
$$\begin{aligned} \dot x &= \frac yx\\ \dot y &= \left( \frac{1-x}{x} \right) y \end{aligned}$$
What I did was the following. From the 1st ODE, assuming $x \neq 0$, I got $y = x \dot x$ and, hence,
$$y = \frac 12 \frac{\mathrm d}{\mathrm d t} \left(x^2 \right)$$
and, plugging into the 2nd ODE,
$$\frac{\mathrm d^2}{\mathrm d t^2} \left(x^2 \right) = \left( \frac{1-x}{x} \right) \frac{\mathrm d}{\mathrm d t} \left(x^2 \right)$$
which eventually yields the following nonlinear 2nd order ODE
$$x \ddot x + \dot x \left( \dot x + x – 1 \right) = 0$$
which Wolfram Alpha can solve. However, this looks messy. I am looking for cleaner solutions.
Best Answer
From the quotient of both equations you get $$ \frac{dy}{dx}=1-x $$ so that $y=C+x-\frac12x^2$ and then $$ \dot x=\frac{C}x+1-\frac12x $$ which can be solved as separable DE.