Solve the differential equation √(1-x^2) dy+√(1-y^2) dx=0

Solve the differential equation $\sqrt{1-x^2}\,dy+\sqrt{1-y^2}\,dx=0$.

Explanation of Method

Separate variables into two standard $\arcsin$ integrals, requiring $|x|<1$ and $|y|<1$.

Step 1: Separate variables

\[\frac{dy}{\sqrt{1-y^2}}=-\frac{dx}{\sqrt{1-x^2}},\qquad |x|<1,\ |y|<1.\]

Step 2: Integrate

\[\arcsin y=-\arcsin x+C \quad\Longrightarrow\quad \boxed{\,\arcsin x+\arcsin y=C.}\]

Step 3: Equivalent explicit form

Using $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$ with $\alpha=\arcsin x,\ \beta=\arcsin y$, and $\cos(\arcsin u)=\sqrt{1-u^2}$, the relation is equivalent to

\[y\sqrt{1-x^2}+x\sqrt{1-y^2}=k,\qquad |k|\le1.\]

Result

Final answer:

\[\boxed{\,\arcsin x+\arcsin y=C\ \ \Longleftrightarrow\ \ y\sqrt{1-x^2}+x\sqrt{1-y^2}=k.}\]