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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}d=-\frac{y-x}{x\left(2x+3y\right)}\text{, }&x\neq -\frac{3y}{2}\text{ and }x\neq 0\\d\in \mathrm{C}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

Subtract y from both sides. Anything subtracted from zero gives its negation.

$$2dx^{2}+3ydx-x=-y$$

Add x to both sides.

$$2dx^{2}+3ydx=-y+x$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide both sides by 2x^{2}+3xy.

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

Dividing by 2x^{2}+3xy undoes the multiplication by 2x^{2}+3xy.

$$d=\frac{x-y}{2x^{2}+3xy}$$

Divide x-y by 2x^{2}+3xy.

$$d=\frac{x-y}{x\left(2x+3y\right)}$$

Solve for d

$$\left\{\begin{matrix}d=-\frac{y-x}{x\left(2x+3y\right)}\text{, }&x\neq -\frac{3y}{2}\text{ and }x\neq 0\\d\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

Subtract y from both sides. Anything subtracted from zero gives its negation.

$$2dx^{2}+3ydx-x=-y$$

Add x to both sides.

$$2dx^{2}+3ydx=-y+x$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide both sides by 2x^{2}+3xy.

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

Dividing by 2x^{2}+3xy undoes the multiplication by 2x^{2}+3xy.

$$d=\frac{x-y}{2x^{2}+3xy}$$

Divide x-y by 2x^{2}+3xy.

$$d=\frac{x-y}{x\left(2x+3y\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{1+9\left(dy\right)^{2}-14dy}-3dy+1}{4d}\text{; }x=\frac{-\sqrt{1+9\left(dy\right)^{2}-14dy}-3dy+1}{4d}\text{, }&d\neq 0\\x=y\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{1+9\left(dy\right)^{2}-14dy}-3dy+1}{4d}\text{; }x=\frac{-\sqrt{1+9\left(dy\right)^{2}-14dy}-3dy+1}{4d}\text{, }&\left(y\geq \frac{4\sqrt{10}|d|+14d}{18d^{2}}\text{ or }y\leq -\frac{4\sqrt{10}|d|-14d}{18d^{2}}\right)\text{ and }d\neq 0\\x=y\text{, }&d=0\end{matrix}\right.$$