Solve vdx+(x+y^2)dy=0

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

Solve for d

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

Steps for Solving Linear Equation

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

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

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

$$vdx+xdy+dy^{3}=0$$

Combine all terms containing d.

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

Divide 0 by vx+xy+y^{3}.

$$d=0$$

Solve for v

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

Steps for Solving Linear Equation

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

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

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

$$vdx+xdy+dy^{3}=0$$

Subtract xdy from both sides. Anything subtracted from zero gives its negation.

$$vdx+dy^{3}=-xdy$$

Subtract dy^{3} from both sides.

$$vdx=-xdy-dy^{3}$$

Reorder the terms.

$$dvx=-dxy-dy^{3}$$

The equation is in standard form.

$$dxv=-dxy-dy^{3}$$

Divide both sides by dx.

$$\frac{dxv}{dx}=-\frac{dy\left(x+y^{2}\right)}{dx}$$

Dividing by dx undoes the multiplication by dx.

$$v=-\frac{dy\left(x+y^{2}\right)}{dx}$$

Divide -dy\left(x+y^{2}\right) by dx.

$$v=-\frac{y\left(x+y^{2}\right)}{x}$$