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

$$( x + 2 y ) d x + ( 2 x + 3 y ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-3y\text{ or }x=-y\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(xd+2yd\right)x+\left(2x+3y\right)dy=0$$

Use the distributive property to multiply xd+2yd by x.

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

Use the distributive property to multiply 2x+3y by d.

$$dx^{2}+2ydx+\left(2xd+3yd\right)y=0$$

Use the distributive property to multiply 2xd+3yd by y.

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

Combine 2ydx and 2xdy to get 4ydx.

$$dx^{2}+4ydx+3dy^{2}=0$$

Combine all terms containing d.

$$\left(x^{2}+4yx+3y^{2}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}+4xy+3y^{2}\right)d=0$$

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=-3y\text{ or }x=-y\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(xd+2yd\right)x+\left(2x+3y\right)dy=0$$

Use the distributive property to multiply xd+2yd by x.

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

Use the distributive property to multiply 2x+3y by d.

$$dx^{2}+2ydx+\left(2xd+3yd\right)y=0$$

Use the distributive property to multiply 2xd+3yd by y.

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

Combine 2ydx and 2xdy to get 4ydx.

$$dx^{2}+4ydx+3dy^{2}=0$$

Combine all terms containing d.

$$\left(x^{2}+4yx+3y^{2}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}+4xy+3y^{2}\right)d=0$$

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-3y\text{; }x=-y\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=-3y\text{; }x=-y\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$