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

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

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

$$ydx+xdy-dy^{3}=0$$

Combine ydx and xdy to get 2ydx.

$$2ydx-dy^{3}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by 2yx-y^{3}.

$$d=0$$

Solve for x (complex solution)

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

Steps for Solving Linear Equation

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

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

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

$$ydx+xdy-dy^{3}=0$$

Combine ydx and xdy to get 2ydx.

$$2ydx-dy^{3}=0$$

Add dy^{3} to both sides. Anything plus zero gives itself.

$$2ydx=dy^{3}$$

The equation is in standard form.

$$2dyx=dy^{3}$$

Divide both sides by 2yd.

$$\frac{2dyx}{2dy}=\frac{dy^{3}}{2dy}$$

Dividing by 2yd undoes the multiplication by 2yd.

$$x=\frac{dy^{3}}{2dy}$$

Divide dy^{3} by 2yd.

$$x=\frac{y^{2}}{2}$$

Solve for d

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

Steps for Solving Linear Equation

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

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

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

$$ydx+xdy-dy^{3}=0$$

Combine ydx and xdy to get 2ydx.

$$2ydx-dy^{3}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by 2yx-y^{3}.

$$d=0$$

Solve for x

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

Steps for Solving Linear Equation

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

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

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

$$ydx+xdy-dy^{3}=0$$

Combine ydx and xdy to get 2ydx.

$$2ydx-dy^{3}=0$$

Add dy^{3} to both sides. Anything plus zero gives itself.

$$2ydx=dy^{3}$$

The equation is in standard form.

$$2dyx=dy^{3}$$

Divide both sides by 2yd.

$$\frac{2dyx}{2dy}=\frac{dy^{3}}{2dy}$$

Dividing by 2yd undoes the multiplication by 2yd.

$$x=\frac{dy^{3}}{2dy}$$

Divide dy^{3} by 2yd.

$$x=\frac{y^{2}}{2}$$