solve two-dimensional nonlinear system numerically

Question

Given the system $$x' = x(1 − y^2) \\ y' = y^2(1 − x^2)$$

I can plot the trajectories according to the analysis of its fixed points, but I need help with solving it numerically.

Can anyone help me with solving this system numerically?

Answer 1

$$\frac{dx}{dt}=x(1-y^2)$$ $$\frac{dy}{dt}=y^2(1-x^2)$$ $$\frac{dy}{dx}=\frac{y^2(1-x^2)}{x(1-y^2)}$$ $$\frac{1-y^2}{y^2}dy=\frac{1-x^2}{x}dx$$ After integration: $$-\frac{1}{y}-y=\ln|x|-\frac{x^2}{2}+c$$ $$y^2+(\ln|x|-\frac{x^2}{2}+c)y+1=0$$ $$y=\frac{ -(\ln|x|-\frac{x^2}{2}+c) \pm \sqrt{(\ln|x|-\frac{x^2}{2}+c)^2-4}}{2}$$ Then you can compute $y(x)$ numerically.

Example with numerical method :

Starting point : $(t=0,x=2,y=3)$

Final point : $t=1/2,x\simeq 0.167,y\simeq 3.55)$

Without computer, it is possible to compute approximates "by hand", but with lower accuracy.