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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=0\text{ or }\left(y=-\frac{x+2}{2x\left(x-1\right)}\text{ and }x\neq 1\text{ and }x\neq 0\right)\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply xy^{2}+2x^{2}y^{3} by d.

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

Use the distributive property to multiply xy^{2}d+2x^{2}y^{3}d by x.

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

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

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

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

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

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

$$y^{2}dx^{2}+2y^{3}dx^{3}+2xdy^{2}-2x^{2}dy^{3}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\text{ or }\left(y=-\frac{x+2}{2x\left(x-1\right)}\text{ and }x\neq 1\text{ and }x\neq 0\right)\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply xy^{2}+2x^{2}y^{3} by d.

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

Use the distributive property to multiply xy^{2}d+2x^{2}y^{3}d by x.

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

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

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

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

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

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

$$y^{2}dx^{2}+2y^{3}dx^{3}+2xdy^{2}-2x^{2}dy^{3}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{4y^{2}-20y+1}+2y-1}{4y}\text{; }x=\frac{-\sqrt{4y^{2}-20y+1}+2y-1}{4y}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{4y^{2}-20y+1}+2y-1}{4y}\text{; }x=\frac{-\sqrt{4y^{2}-20y+1}+2y-1}{4y}\text{, }&\left(y\neq 0\text{ and }y\leq \frac{5}{2}-\sqrt{6}\right)\text{ or }y\geq \sqrt{6}+\frac{5}{2}\\x\in \mathrm{R}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$