Assume solutions of the equation
![\[ y \sqrt{1-x^2} y' = x \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-2b0eb8680894edb812888f54d533ce97_l3.png "Rendered by QuickLaTeX.com")
exist and find an implicit formula satisfied by these solutions.
This is a separable first order equation. We compute
![\begin{align*} y \sqrt{1-x^2} y' = x && \implies && yy' &= \frac{x}{\sqrt{1-x^2}} \\[9pt] && \implies && \int y \, dy &= \int \frac{x}{\sqrt{1-x^2}} \, dx \\[9pt] && \implies && \frac{1}{2} y^2 &= -\sqrt{1-x^2} + C \\[9pt] && \implies && y^2 + 2 \sqrt{1-x^2} &= C. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b14ffad489ff7ac6dda8c740df21534a_l3.png "Rendered by QuickLaTeX.com")
Assume solutions of the equation
exist and find an implicit formula satisfied by these solutions.
This is a separable first order equation. We compute