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THE ODEs has the following form: \begin{align*} \frac{dy}{dt} &= - \lambda xy\\ \frac{dx}{dt} &= -\eta y^2, \end{align*} where $\lambda$ and $\eta$ are constants. $y(0) = C_1 >0$ and $x(0) = C_2 >0$.

Is there any standard tool to analyze it? Thanks.

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2 Answers 2

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You can deduce that: $$\dfrac {dy}{dx}=\dfrac {\lambda x}{\eta y}$$ $$\eta y^2 -\lambda x^2=C$$

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    $\begingroup$ Therefore integral curves are conics... $\endgroup$
    – Jean Marie
    Apr 11, 2021 at 23:33
  • $\begingroup$ @JeanMarie Yes thats exact. $\endgroup$ Apr 11, 2021 at 23:55
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If $x$ and $y$ only depend on $t$ i think you could just have

$$y=\pm\sqrt{-\frac{dx}{dt}\frac{1}{\eta}}$$

and substitute it into the first equation.

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