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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&\left(y=-\frac{x^{2}}{x+1}\text{ and }x\neq -1\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

$$dx^{3}+ydx^{2}+xdy=0$$

Combine all terms containing d.

$$\left(x^{3}+yx^{2}+xy\right)d=0$$

The equation is in standard form.

$$\left(x^{3}+xy+yx^{2}\right)d=0$$

Divide 0 by x^{3}+yx^{2}+xy.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(y=-\frac{x^{2}}{x+1}\text{ and }x\neq -1\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

$$dx^{3}+ydx^{2}+xdy=0$$

Combine all terms containing d.

$$\left(x^{3}+yx^{2}+xy\right)d=0$$

The equation is in standard form.

$$\left(x^{3}+xy+yx^{2}\right)d=0$$

Divide 0 by x^{3}+yx^{2}+xy.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{-\sqrt{y\left(y-4\right)}-y}{2}\text{; }x=0\text{; }x=\frac{\sqrt{y\left(y-4\right)}-y}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{y\left(y-4\right)}-y}{2}\text{; }x=\frac{-\sqrt{y\left(y-4\right)}-y}{2}\text{, }&y\geq 4\text{ or }y\leq 0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$