Solve 2(y+x)dx=xy+y

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

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{y\left(x+1\right)}{2x\left(x+y\right)}\text{, }&x\neq 0\text{ and }y\neq -x\\d\in \mathrm{C}\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }\left(x=-1\text{ and }y=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(2y+2x\right)dx=xy+y$$

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

$$\left(2yd+2xd\right)x=xy+y$$

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

$$2ydx+2dx^{2}=xy+y$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

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

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

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

Divide yx+y by 2x^{2}+2xy.

$$d=\frac{y\left(x+1\right)}{2x\left(x+y\right)}$$

Solve for d

$$\left\{\begin{matrix}d=\frac{y\left(x+1\right)}{2x\left(x+y\right)}\text{, }&x\neq 0\text{ and }y\neq -x\\d\in \mathrm{R}\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }\left(x=-1\text{ and }y=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(2y+2x\right)dx=xy+y$$

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

$$\left(2yd+2xd\right)x=xy+y$$

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

$$2ydx+2dx^{2}=xy+y$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

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

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

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

Divide yx+y by 2x^{2}+2xy.

$$d=\frac{y\left(x+1\right)}{2x\left(x+y\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{y\left(y\left(2d-1\right)^{2}+8d\right)}-2dy+y}{4d}\text{; }x=\frac{-\sqrt{y\left(y\left(2d-1\right)^{2}+8d\right)}-2dy+y}{4d}\text{, }&d\neq 0\\x=-1\text{, }&y\neq 0\text{ and }d=0\\x\in \mathrm{C}\text{, }&d=0\text{ and }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{y\left(4yd^{2}-4dy+y+8d\right)}-2dy+y}{4d}\text{; }x=\frac{-\sqrt{y\left(4yd^{2}-4dy+y+8d\right)}-2dy+y}{4d}\text{, }&\left(d\neq 0\text{ and }d\neq \frac{1}{2}\text{ and }y=-\frac{8d}{4d^{2}-4d+1}\right)\text{ or }\left(d\neq 0\text{ and }d\neq \frac{1}{2}\text{ and }y\leq -\frac{8d}{4d^{2}-4d+1}\text{ and }y\leq 0\right)\text{ or }\left(d\neq 0\text{ and }y\geq 0\text{ and }y\geq -\frac{8d}{4d^{2}-4d+1}\right)\text{ or }\left(d=\frac{1}{2}\text{ and }y\geq 0\right)\text{ or }\left(d\neq 0\text{ and }y=0\right)\\x=-1\text{, }&y\neq 0\text{ and }d=0\\x\in \mathrm{R}\text{, }&d=0\text{ and }y=0\end{matrix}\right.$$