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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-5y\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply 2xd-3yd by x.

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

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

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

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

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

To find the opposite of 2ydx+3dx^{2}, find the opposite of each term.

$$2dx^{2}-3ydx-2ydx-3dx^{2}=0$$

Combine -3ydx and -2ydx to get -5ydx.

$$2dx^{2}-5ydx-3dx^{2}=0$$

Combine 2dx^{2} and -3dx^{2} to get -dx^{2}.

$$-dx^{2}-5ydx=0$$

Combine all terms containing d.

$$\left(-x^{2}-5yx\right)d=0$$

The equation is in standard form.

$$\left(-x^{2}-5xy\right)d=0$$

Divide 0 by -x^{2}-5yx.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=-5y\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply 2xd-3yd by x.

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

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

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

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

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

To find the opposite of 2ydx+3dx^{2}, find the opposite of each term.

$$2dx^{2}-3ydx-2ydx-3dx^{2}=0$$

Combine -3ydx and -2ydx to get -5ydx.

$$2dx^{2}-5ydx-3dx^{2}=0$$

Combine 2dx^{2} and -3dx^{2} to get -dx^{2}.

$$-dx^{2}-5ydx=0$$

Combine all terms containing d.

$$\left(-x^{2}-5yx\right)d=0$$

The equation is in standard form.

$$\left(-x^{2}-5xy\right)d=0$$

Divide 0 by -x^{2}-5yx.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-5y\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=-5y\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$