Solve (3xy+y^2)dx+(x^2+xy)dy=0

$$( 3 x y + y ^ { 2 } ) d x + ( x ^ { 2 } + x y ) d y = 0$$

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 3xy+y^{2} by d.

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

Use the distributive property to multiply 3xyd+y^{2}d by x.

$$3ydx^{2}+y^{2}dx+\left(x^{2}+xy\right)dy=0$$

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

$$3ydx^{2}+y^{2}dx+\left(x^{2}d+xyd\right)y=0$$

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

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

Combine 3ydx^{2} and x^{2}dy to get 4ydx^{2}.

$$4ydx^{2}+y^{2}dx+xdy^{2}=0$$

Combine y^{2}dx and xdy^{2} to get 2y^{2}dx.

$$4ydx^{2}+2y^{2}dx=0$$

Combine all terms containing d.

$$\left(4yx^{2}+2y^{2}x\right)d=0$$

The equation is in standard form.

$$\left(2xy^{2}+4yx^{2}\right)d=0$$

Divide 0 by 4yx^{2}+2y^{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 3xy+y^{2} by d.

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

Use the distributive property to multiply 3xyd+y^{2}d by x.

$$3ydx^{2}+y^{2}dx+\left(x^{2}+xy\right)dy=0$$

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

$$3ydx^{2}+y^{2}dx+\left(x^{2}d+xyd\right)y=0$$

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

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

Combine 3ydx^{2} and x^{2}dy to get 4ydx^{2}.

$$4ydx^{2}+y^{2}dx+xdy^{2}=0$$

Combine y^{2}dx and xdy^{2} to get 2y^{2}dx.

$$4ydx^{2}+2y^{2}dx=0$$

Combine all terms containing d.

$$\left(4yx^{2}+2y^{2}x\right)d=0$$

The equation is in standard form.

$$\left(2xy^{2}+4yx^{2}\right)d=0$$

Divide 0 by 4yx^{2}+2y^{2}x.

$$d=0$$

Solve for x (complex solution)

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

Solve for x

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