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I came across this ordinary differential equation $$x \,(y')^2+a=0\,,$$ where $a$ is a real number, $x$ is a real variable and $y$ is a real algebraic function of $x$. I was wondering whether this is a well-known and studied ODE or not.

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2 Answers 2

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If you make the change of variable $x = t^4$, $y(x) = u(t)$, you find $u$ satisfies Chrystal's equation.

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  • $\begingroup$ Thanks a lot for the answer! $\endgroup$
    – Cyclops
    Feb 11, 2021 at 8:56
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If $a<0$ and $y(x):= 2 \sqrt{-a} \sqrt{x}$, then $y$ is a solution of differential equation on $(0, \infty).$

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  • $\begingroup$ thank you for your answer but I was looking for a different one. Maybe I didn't explain well enough in my post. I know what the solutions of this equation are. I was just wondering whether or not it is a special case of a well-known ODE, like the Bernoulli or the Riccati equations. $\endgroup$
    – Cyclops
    Feb 10, 2021 at 12:01

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