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

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply y by x+y.

$$x^{2}dy+\left(yx+y^{2}\right)dx=0$$

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

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

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

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

Combine x^{2}dy and ydx^{2} to get 2x^{2}dy.

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

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by 2x^{2}y+y^{2}x.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

Use the distributive property to multiply y by x+y.

$$x^{2}dy+\left(yx+y^{2}\right)dx=0$$

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

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

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

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

Combine x^{2}dy and ydx^{2} to get 2x^{2}dy.

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

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by 2x^{2}y+y^{2}x.

$$d=0$$

Solve for x (complex solution)

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

Solve for x

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