Solve xydy=(x+y)dx

$$x y d y = ( x + y ) d x$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=y\left(y-1\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply x+y by d.

$$xy^{2}d=\left(xd+yd\right)x$$

Use the distributive property to multiply xd+yd by x.

$$xy^{2}d=dx^{2}+ydx$$

Subtract dx^{2} from both sides.

$$xy^{2}d-dx^{2}=ydx$$

Subtract ydx from both sides.

$$xy^{2}d-dx^{2}-ydx=0$$

Reorder the terms.

$$-dx^{2}+dxy^{2}-dxy=0$$

Combine all terms containing d.

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

The equation is in standard form.

$$\left(xy^{2}-x^{2}-xy\right)d=0$$

Divide 0 by -x^{2}+xy^{2}-xy.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=y\left(y-1\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply x+y by d.

$$xy^{2}d=\left(xd+yd\right)x$$

Use the distributive property to multiply xd+yd by x.

$$xy^{2}d=dx^{2}+ydx$$

Subtract dx^{2} from both sides.

$$xy^{2}d-dx^{2}=ydx$$

Subtract ydx from both sides.

$$xy^{2}d-dx^{2}-ydx=0$$

Reorder the terms.

$$-dx^{2}+dxy^{2}-dxy=0$$

Combine all terms containing d.

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

The equation is in standard form.

$$\left(xy^{2}-x^{2}-xy\right)d=0$$

Divide 0 by -x^{2}+xy^{2}-xy.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=y\left(y-1\right)\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=y\left(y-1\right)\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$