Solve xdx+y*dy+xdy-ydx/x^2+y^2=0

$$x d x + y \cdot d y + \frac { x d y - y d x } { x ^ { 2 } + y ^ { 2 } } = 0$$

Solve for d (complex solution)

$$d=0$$

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

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-iy\right)\left(x+iy\right).

$$xdx\left(x-iy\right)\left(x+iy\right)+ydy\left(x-iy\right)\left(x+iy\right)+xdy-ydx=0$$

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

$$x^{2}d\left(x-iy\right)\left(x+iy\right)+ydy\left(x-iy\right)\left(x+iy\right)+xdy-ydx=0$$

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

$$x^{2}d\left(x-iy\right)\left(x+iy\right)+y^{2}d\left(x-iy\right)\left(x+iy\right)+xdy-ydx=0$$

Use the distributive property to multiply x^{2}d by x-iy.

$$\left(dx^{3}-ix^{2}dy\right)\left(x+iy\right)+y^{2}d\left(x-iy\right)\left(x+iy\right)+xdy-ydx=0$$

Use the distributive property to multiply dx^{3}-ix^{2}dy by x+iy and combine like terms.

$$dx^{4}+dx^{2}y^{2}+y^{2}d\left(x-iy\right)\left(x+iy\right)+xdy-ydx=0$$

Use the distributive property to multiply y^{2}d by x-iy.

$$dx^{4}+dx^{2}y^{2}+\left(y^{2}dx-idy^{3}\right)\left(x+iy\right)+xdy-ydx=0$$

Use the distributive property to multiply y^{2}dx-idy^{3} by x+iy and combine like terms.

$$dx^{4}+dx^{2}y^{2}+y^{2}dx^{2}+dy^{4}+xdy-ydx=0$$

Combine dx^{2}y^{2} and y^{2}dx^{2} to get 2dx^{2}y^{2}.

$$dx^{4}+2dx^{2}y^{2}+dy^{4}+xdy-ydx=0$$

Combine xdy and -ydx to get 0.

$$dx^{4}+2dx^{2}y^{2}+dy^{4}+0=0$$

Anything plus zero gives itself.

$$dx^{4}+2dx^{2}y^{2}+dy^{4}=0$$

Combine all terms containing d.

$$\left(x^{4}+2x^{2}y^{2}+y^{4}\right)d=0$$

Divide 0 by x^{4}+2x^{2}y^{2}+y^{4}.

$$d=0$$

Solve for d

$$d=0$$

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}+y^{2}.

$$xdx\left(x^{2}+y^{2}\right)+ydy\left(x^{2}+y^{2}\right)+xdy-ydx=0$$

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

$$x^{2}d\left(x^{2}+y^{2}\right)+ydy\left(x^{2}+y^{2}\right)+xdy-ydx=0$$

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

$$x^{2}d\left(x^{2}+y^{2}\right)+y^{2}d\left(x^{2}+y^{2}\right)+xdy-ydx=0$$

Use the distributive property to multiply x^{2}d by x^{2}+y^{2}.

$$dx^{4}+x^{2}dy^{2}+y^{2}d\left(x^{2}+y^{2}\right)+xdy-ydx=0$$

Use the distributive property to multiply y^{2}d by x^{2}+y^{2}.

$$dx^{4}+x^{2}dy^{2}+y^{2}dx^{2}+dy^{4}+xdy-ydx=0$$

Combine x^{2}dy^{2} and y^{2}dx^{2} to get 2x^{2}dy^{2}.

$$dx^{4}+2x^{2}dy^{2}+dy^{4}+xdy-ydx=0$$

Combine xdy and -ydx to get 0.

$$dx^{4}+2x^{2}dy^{2}+dy^{4}+0=0$$

Anything plus zero gives itself.

$$dx^{4}+2x^{2}dy^{2}+dy^{4}=0$$

Combine all terms containing d.

$$\left(x^{4}+2x^{2}y^{2}+y^{4}\right)d=0$$

Divide 0 by x^{4}+2x^{2}y^{2}+y^{4}.

$$d=0$$

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus iy,-iy$$

$$d=0$$

Solve for x

$$\left\{\begin{matrix}x\in \mathrm{R}\text{, }&d=0\text{ and }y\neq 0\\x\neq 0\text{, }&d=0\end{matrix}\right.$$