Solve sqrt{1-y^2}du+sqrt{1-u^2}dy=0

$$\sqrt { 1 - y ^ { 2 } } d u + \sqrt { 1 - u ^ { 2 } } d y = 0$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&|y|\leq 1\text{ and }|u|\leq 1\\d\in \mathrm{R}\text{, }&\left(y=-u\text{ and }u\geq 0\text{ and }u\leq 1\right)\text{ or }\left(y=-u\text{ and }u\leq 0\text{ and }u\geq -1\right)\text{ or }\left(y=-1\text{ and }u=-1\right)\text{ or }\left(y=1\text{ and }u=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Combine all terms containing d.

$$\left(\sqrt{1-y^{2}}u+\sqrt{1-u^{2}}y\right)d=0$$

The equation is in standard form.

$$\left(u\sqrt{1-y^{2}}+y\sqrt{1-u^{2}}\right)d=0$$

Divide 0 by \sqrt{1-y^{2}}u+\sqrt{1-u^{2}}y.

$$d=0$$

Solve for u

$$\left\{\begin{matrix}u=-y\text{, }&|y|\leq 1\\u=y\text{, }&|y|=1\text{ or }y=0\\u\in \begin{bmatrix}-1,1\end{bmatrix}\text{, }&d=0\text{ and }|y|\leq 1\end{matrix}\right.$$