Let y=y(x) solve √1-x^2dydx+√1-y^2=0, |x|

Question

Let $y=y(x)$ solve $\sqrt{1-x^2}\dfrac{dy}{dx}+\sqrt{1-y^2}=0$, $|x|<1$. If $y\left(\dfrac12\right)=\dfrac{\sqrt3}{2}$, then $y\left(-\dfrac{1}{\sqrt2}\right)$ is:

Accepted Answer

$\sin^{-1}y+\sin^{-1}x=c$. Using $x=1/2,y=\sqrt3/2$: $c=\pi/2$, so $\sin^{-1}y=\cos^{-1}x$.

At $x=-1/\sqrt2$: $y=\sin(\cos^{-1}(-1/\sqrt2)) = \sin(3\pi/4)=\dfrac{1}{\sqrt2}$.

Solutions · 1