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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&y=-x^{3}\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

Combine ydx and -2xdy to get -ydx.

$$-ydx-dx^{4}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by -yx-x^{4}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y=-x^{3}\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

Combine ydx and -2xdy to get -ydx.

$$-ydx-dx^{4}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by -yx-x^{4}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=e^{\frac{\pi i}{3}}\sqrt[3]{y}\text{; }x=0\text{; }x=-\sqrt[3]{y}\text{; }x=e^{\frac{5\pi i}{3}}\sqrt[3]{y}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=-\sqrt[3]{y}\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$