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

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

To find the opposite of x^{2}dy^{2}+dy^{2}, find the opposite of each term.

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

Combine y^{2}dx^{2} and -x^{2}dy^{2} to get 0.

$$dx^{2}-dy^{2}=0$$

Combine all terms containing d.

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

Divide 0 by x^{2}-y^{2}.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

To find the opposite of x^{2}dy^{2}+dy^{2}, find the opposite of each term.

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

Combine y^{2}dx^{2} and -x^{2}dy^{2} to get 0.

$$dx^{2}-dy^{2}=0$$

Combine all terms containing d.

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

Divide 0 by x^{2}-y^{2}.

$$d=0$$

Solve for x (complex solution)

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

Solve for x

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