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

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

Resolver d (complex solution)

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

Pasos para resolver unha ecuación lineal

Usa a propiedade distributiva para multiplicar 2 por y+x.

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

Usa a propiedade distributiva para multiplicar 2y+2x por d.

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

Usa a propiedade distributiva para multiplicar 2yd+2xd por x.

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

Combina todos os termos que conteñan d.

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

A ecuación está en forma estándar.

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

Divide ambos lados entre 2x^{2}+2xy.

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

A división entre 2x^{2}+2xy desfai a multiplicación por 2x^{2}+2xy.

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

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

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

Resolver d

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

Pasos para resolver unha ecuación lineal

Usa a propiedade distributiva para multiplicar 2 por y+x.

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

Usa a propiedade distributiva para multiplicar 2y+2x por d.

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

Usa a propiedade distributiva para multiplicar 2yd+2xd por x.

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

Combina todos os termos que conteñan d.

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

A ecuación está en forma estándar.

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

Divide ambos lados entre 2x^{2}+2xy.

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

A división entre 2x^{2}+2xy desfai a multiplicación por 2x^{2}+2xy.

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

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

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

Resolver x (complex solution)

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

Resolver x

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