Find solution of this ODE

Question

I try to solve this ODE
$$
y^2left(1+left(dfrac{dy}{dx}right)^2right)=C
$$
where $C$ is positive constant.
Then I have $y=pmsqrt{C}$.
Is it right $y=pmsqrt{C}$ is only the solution to that ODE?

Math Lover · Answer

$y^2left(1+left(dfrac{dy}{dx}right)^2right)=C$
$dfrac{dy}{dx} = pm sqrt {frac{C}{y^2} - 1} ,$ (for $|y| lt C$)
$dfrac{ydy}{sqrt{C-y^2}} = pm dx$
$x = C' pm {sqrt{C-y^2}} implies (x - C')^2 + y^2 = C$

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