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

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

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

$$x^{2}+ydx^{2}+y^{2}dx=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$ydx^{2}+y^{2}dx=-x^{2}$$

Combine all terms containing d.

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

The equation is in standard form.

$$\left(xy^{2}+yx^{2}\right)d=-x^{2}$$

Divide both sides by yx^{2}+y^{2}x.

$$\frac{\left(xy^{2}+yx^{2}\right)d}{xy^{2}+yx^{2}}=-\frac{x^{2}}{xy^{2}+yx^{2}}$$

Dividing by yx^{2}+y^{2}x undoes the multiplication by yx^{2}+y^{2}x.

$$d=-\frac{x^{2}}{xy^{2}+yx^{2}}$$

Divide -x^{2} by yx^{2}+y^{2}x.

$$d=-\frac{x}{y\left(x+y\right)}$$

Solve for d

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

Steps for Solving Linear Equation

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

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

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

$$x^{2}+ydx^{2}+y^{2}dx=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$ydx^{2}+y^{2}dx=-x^{2}$$

Combine all terms containing d.

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

The equation is in standard form.

$$\left(xy^{2}+yx^{2}\right)d=-x^{2}$$

Divide both sides by yx^{2}+y^{2}x.

$$\frac{\left(xy^{2}+yx^{2}\right)d}{xy^{2}+yx^{2}}=-\frac{x^{2}}{xy^{2}+yx^{2}}$$

Dividing by yx^{2}+y^{2}x undoes the multiplication by yx^{2}+y^{2}x.

$$d=-\frac{x^{2}}{xy^{2}+yx^{2}}$$

Divide -x^{2} by yx^{2}+y^{2}x.

$$d=-\frac{x}{y\left(x+y\right)}$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=-\frac{dy^{2}}{dy+1}\text{, }&d=0\text{ or }y\neq -\frac{1}{d}\end{matrix}\right.$$