A solution of the differential equation \(\sqrt{1-x^{2}}dy+\sqrt{1-x^{2}}dx=0 (|x|<1,|y|<1)\) is

\(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}=c\) \(x\sin^{-1}y+y\sin^{-1}x=c\) \(\frac{x^{2}}{\sqrt{1-x^{2}}}+\frac{y^{2}}{\sqrt{1-y^{2}}}=c\) \(x\sqrt{1-x^{2}}+y\sqrt{1-y^{2}}=c\)

\(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}=c\)

\(\begin{aligned}\text{Given, }\sqrt{1-x^{2}}dy&=-\sqrt{1-y^{2}}dx\\ \sin^{-1}y+\sin^{-1}x&=\sin^{-1}c\\ \sin^{-1}[x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}]&=\sin^{-1}c\end{aligned}\)