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I have the following ODE system and have been tasked to find all of it's orbits.

The system is $$x' = y(x^2 + 1)$$ $$y' = x(x^2 + 1)$$

From here is where I'm lost. I found one definition of an orbit that states:

"if $x(t)$ = is a solution to $x' = f(x),$ with maximal interval $I$, then the set {$x(t)$ for $t$ in $I$} a subset of the domain, is the orbit of the equation"

but I don't understand for the given system how to solve for this.

I tried solving the system for both equations and got: $$x = \tan\left(\frac12 \cdot y^2 + c_1\right)$$ $$y = \left(\frac14 \cdot x^4 + \frac12 \cdot x^2 + c_2\right)$$ but from here I'm confused on how to extract the orbits of the equations.

I've also tried looking around at other MSE questions relating to orbits but I'm still lost. Any help or direction would be great, thanks!

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$$x' = y(x^2 + 1)$$ $$y' = x(x^2 + 1)$$ Rewrite this as: $$\implies \dfrac {dx}{dy}=\dfrac {y(x^2+1)}{x(x^2+1)}$$ $$x{dx}=y{dy}$$ You deduce all the orbits: $$x^2-y^2=C$$

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  • $\begingroup$ Ah I see. So is it always that easy to solve for the orbits? You just put dx/dy and solve? how'd you know to do that? $\endgroup$ Commented Sep 24, 2020 at 23:32
  • $\begingroup$ Sometimes it's as easy as that. Sometimes it's not. it depends on the system of DE"s @BigMeech420 $\endgroup$ Commented Sep 24, 2020 at 23:36
  • $\begingroup$ The orbits are hyperbolas @BigMeech420 $\endgroup$ Commented Sep 24, 2020 at 23:51

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