Solve xdx+ydy=2

$$x d x + y d y = 2$$

Solve for d (complex solution)

$$d=\frac{2}{x^{2}+y^{2}}$$

$$x\neq iy\text{ and }x\neq -iy$$

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

$$x^{2}d+ydy=2$$

Multiply y and y to get y^{2}.

$$x^{2}d+y^{2}d=2$$

Combine all terms containing d.

$$\left(x^{2}+y^{2}\right)d=2$$

Divide both sides by x^{2}+y^{2}.

$$\frac{\left(x^{2}+y^{2}\right)d}{x^{2}+y^{2}}=\frac{2}{x^{2}+y^{2}}$$

Dividing by x^{2}+y^{2} undoes the multiplication by x^{2}+y^{2}.

$$d=\frac{2}{x^{2}+y^{2}}$$

Solve for d

$$d=\frac{2}{x^{2}+y^{2}}$$

$$y\neq 0\text{ or }x\neq 0$$

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

$$x^{2}d+ydy=2$$

Multiply y and y to get y^{2}.

$$x^{2}d+y^{2}d=2$$

Combine all terms containing d.

$$\left(x^{2}+y^{2}\right)d=2$$

Divide both sides by x^{2}+y^{2}.

$$\frac{\left(x^{2}+y^{2}\right)d}{x^{2}+y^{2}}=\frac{2}{x^{2}+y^{2}}$$

Dividing by x^{2}+y^{2} undoes the multiplication by x^{2}+y^{2}.

$$d=\frac{2}{x^{2}+y^{2}}$$

Solve for x (complex solution)

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

$$x=\sqrt{-y^{2}+\frac{2}{d}}\text{, }d\neq 0$$

Solve for x

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

$$x=-\sqrt{-y^{2}+\frac{2}{d}}\text{, }\left(d\leq \frac{2}{y^{2}}\text{ or }y=0\right)\text{ and }d>0$$