Solve y-ydx=sqrt{x^2-y^2}dx

$$y - y d x = \sqrt { x ^ { 2 } - y ^ { 2 } } d x$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{y}{x\left(\sqrt{x^{2}-y^{2}}+y\right)}\text{, }&x\neq 0\text{ and }\left(y\neq -\frac{\sqrt{2}x}{2}\text{ or }arg(x)\geq \pi \right)\text{ and }\left(y\neq -\frac{\sqrt{2}x}{2}\text{ or }y\neq \frac{\sqrt{2}x}{2}\right)\text{ and }\left(arg(x)<\pi \text{ or }y\neq \frac{\sqrt{2}x}{2}\right)\\d\in \mathrm{C}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \sqrt{x^{2}-y^{2}}dx from both sides.

$$y-ydx-\sqrt{x^{2}-y^{2}}dx=0$$

Subtract y from both sides. Anything subtracted from zero gives its negation.

$$-ydx-\sqrt{x^{2}-y^{2}}dx=-y$$

Combine all terms containing d.

$$\left(-yx-\sqrt{x^{2}-y^{2}}x\right)d=-y$$

The equation is in standard form.

$$\left(-x\sqrt{x^{2}-y^{2}}-xy\right)d=-y$$

Divide both sides by -yx-\sqrt{x^{2}-y^{2}}x.

$$\frac{\left(-x\sqrt{x^{2}-y^{2}}-xy\right)d}{-x\sqrt{x^{2}-y^{2}}-xy}=-\frac{y}{-x\sqrt{x^{2}-y^{2}}-xy}$$

Dividing by -yx-\sqrt{x^{2}-y^{2}}x undoes the multiplication by -yx-\sqrt{x^{2}-y^{2}}x.

$$d=-\frac{y}{-x\sqrt{x^{2}-y^{2}}-xy}$$

Divide -y by -yx-\sqrt{x^{2}-y^{2}}x.

$$d=\frac{y}{x\left(\sqrt{\left(x-y\right)\left(x+y\right)}+y\right)}$$

Solve for d

$$\left\{\begin{matrix}d=\frac{y}{x\left(\sqrt{x^{2}-y^{2}}+y\right)}\text{, }&\left(x\neq 0\text{ and }y\geq -|x|\text{ and }|x|=|y|\right)\text{ or }\left(y\neq -\frac{\sqrt{2}x}{2}\text{ and }y\leq x\text{ and }y\geq -x\right)\text{ or }\left(y\neq \frac{\sqrt{2}x}{2}\text{ and }y\leq -x\text{ and }y\geq x\right)\\d\in \mathrm{R}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \sqrt{x^{2}-y^{2}}dx from both sides.

$$y-ydx-\sqrt{x^{2}-y^{2}}dx=0$$

Subtract y from both sides. Anything subtracted from zero gives its negation.

$$-ydx-\sqrt{x^{2}-y^{2}}dx=-y$$

Combine all terms containing d.

$$\left(-yx-\sqrt{x^{2}-y^{2}}x\right)d=-y$$

The equation is in standard form.

$$\left(-x\sqrt{x^{2}-y^{2}}-xy\right)d=-y$$

Divide both sides by -yx-\sqrt{x^{2}-y^{2}}x.

$$\frac{\left(-x\sqrt{x^{2}-y^{2}}-xy\right)d}{-x\sqrt{x^{2}-y^{2}}-xy}=-\frac{y}{-x\sqrt{x^{2}-y^{2}}-xy}$$

Dividing by -yx-\sqrt{x^{2}-y^{2}}x undoes the multiplication by -yx-\sqrt{x^{2}-y^{2}}x.

$$d=-\frac{y}{-x\sqrt{x^{2}-y^{2}}-xy}$$

Divide -y by -yx-\sqrt{x^{2}-y^{2}}x.

$$d=\frac{y}{x\left(\sqrt{\left(x-y\right)\left(x+y\right)}+y\right)}$$