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I have a system of two equations with

$x'=x^2+y^2-5$

$y'=x^2+2y^2-9$

My goal is to find the equilibrium points.

I got $(\sqrt{5-y^2},\pm2)$ , $\left(\pm\sqrt{\dfrac{9}{11}},\sqrt{5-x^2}\right)$ , $(\sqrt{9-2y^2},\pm2)$ and $\left(\pm\sqrt{\dfrac{11}{3}},\dfrac{\sqrt{9-x^2}}{2}\right)$

But I’m not sure whether I did these correctly and would love some feedback.

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2 Answers 2

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This is a linear system of ODEs, so you can treat them as a system of equations. In this case, the system is simple enough that we do not need to use linear algebra. We can simply solve (1) $x^2+y^2=5$ and (2) $x^2+2y^2=9$

Note that $x={-1,1}$ and $y={-2,2}$ solve (1) and you can plug them into (2) to verify, so your steady-state solutions are $p_1=(-1,-2), p_2=(-1,2), p_3=(1,-2), p_4=(1,2)$

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You have

$$x^2 + y^2 = 5$$

$$x^2 + 2y^2 = 9$$

Subtract one from the other and you get $y = \pm 2$. Plug that back in, and you get $(\pm1, \pm 2)$ to be your steady states.

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