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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=0\text{ or }y=-\sqrt{2}x\text{ or }y=\sqrt{2}x\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

$$ydx-2dx^{4}-\left(x-x^{2}y\right)dy=0$$

Use the distributive property to multiply -1 by x-x^{2}y.

$$ydx-2dx^{4}+\left(-x+x^{2}y\right)dy=0$$

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

$$ydx-2dx^{4}+\left(-xd+x^{2}yd\right)y=0$$

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

$$ydx-2dx^{4}-xdy+x^{2}dy^{2}=0$$

Combine ydx and -xdy to get 0.

$$-2dx^{4}+x^{2}dy^{2}=0$$

Combine all terms containing d.

$$\left(-2x^{4}+x^{2}y^{2}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{2}-2x^{4}\right)d=0$$

Divide 0 by -2x^{4}+x^{2}y^{2}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\text{ or }|y|=\sqrt{2}|x|\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

$$ydx-2dx^{4}-\left(x-x^{2}y\right)dy=0$$

Use the distributive property to multiply -1 by x-x^{2}y.

$$ydx-2dx^{4}+\left(-x+x^{2}y\right)dy=0$$

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

$$ydx-2dx^{4}+\left(-xd+x^{2}yd\right)y=0$$

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

$$ydx-2dx^{4}-xdy+x^{2}dy^{2}=0$$

Combine ydx and -xdy to get 0.

$$-2dx^{4}+x^{2}dy^{2}=0$$

Combine all terms containing d.

$$\left(-2x^{4}+x^{2}y^{2}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{2}-2x^{4}\right)d=0$$

Divide 0 by -2x^{4}+x^{2}y^{2}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-\frac{\sqrt{2}y}{2}\text{; }x=0\text{; }x=\frac{\sqrt{2}y}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{\sqrt{2}y}{2}\text{; }x=0\text{; }x=-\frac{\sqrt{2}y}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$