Solve (x+xy^2)dx-(x^2y+y)dx=0

$$( x + x y ^ { 2 } ) d x - ( x ^ { 2 } y + y ) d x = 0$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

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

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

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

$$dx^{2}+y^{2}dx^{2}-\left(x^{2}yd+yd\right)x=0$$

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

$$dx^{2}+y^{2}dx^{2}-\left(ydx^{3}+ydx\right)=0$$

To find the opposite of ydx^{3}+ydx, find the opposite of each term.

$$dx^{2}+y^{2}dx^{2}-ydx^{3}-ydx=0$$

Combine all terms containing d.

$$\left(x^{2}+y^{2}x^{2}-yx^{3}-yx\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{2}+x^{2}-xy-yx^{3}\right)d=0$$

Divide 0 by x^{2}+y^{2}x^{2}-yx^{3}-yx.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

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

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

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

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

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

$$dx^{2}+y^{2}dx^{2}-\left(x^{2}yd+yd\right)x=0$$

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

$$dx^{2}+y^{2}dx^{2}-\left(ydx^{3}+ydx\right)=0$$

To find the opposite of ydx^{3}+ydx, find the opposite of each term.

$$dx^{2}+y^{2}dx^{2}-ydx^{3}-ydx=0$$

Combine all terms containing d.

$$\left(x^{2}+y^{2}x^{2}-yx^{3}-yx\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{2}+x^{2}-xy-yx^{3}\right)d=0$$

Divide 0 by x^{2}+y^{2}x^{2}-yx^{3}-yx.

$$d=0$$

Solve for x (complex solution)

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

Solve for x

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