Solve (y^2+2x^2y)dx+(2x^3-xy)dy=0

$$( y ^ { 2 } + 2 x ^ { 2 } y ) d x + ( 2 x ^ { 3 } - x y ) d y = 0$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

Combine 2ydx^{3} and 2x^{3}dy to get 4ydx^{3}.

$$y^{2}dx+4ydx^{3}-xdy^{2}=0$$

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

$$4ydx^{3}=0$$

The equation is in standard form.

$$4yx^{3}d=0$$

Divide 0 by 4yx^{3}.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

Combine 2ydx^{3} and 2x^{3}dy to get 4ydx^{3}.

$$y^{2}dx+4ydx^{3}-xdy^{2}=0$$

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

$$4ydx^{3}=0$$

The equation is in standard form.

$$4yx^{3}d=0$$

Divide 0 by 4yx^{3}.

$$d=0$$

Solve for x (complex solution)

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

Solve for x

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