Solve 2(1+y)dx-(1+x)dy=0

$$2 ( 1 + y ) d x - ( 1 + x ) d y = 0$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply 2 by 1+y.

$$\left(2+2y\right)dx-\left(1+x\right)dy=0$$

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

$$\left(2d+2yd\right)x-\left(1+x\right)dy=0$$

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

$$2dx+2ydx-\left(1+x\right)dy=0$$

Use the distributive property to multiply 1+x by d.

$$2dx+2ydx-\left(d+xd\right)y=0$$

Use the distributive property to multiply d+xd by y.

$$2dx+2ydx-\left(dy+xdy\right)=0$$

To find the opposite of dy+xdy, find the opposite of each term.

$$2dx+2ydx-dy-xdy=0$$

Combine 2ydx and -xdy to get ydx.

$$2dx+ydx-dy=0$$

Combine all terms containing d.

$$\left(2x+yx-y\right)d=0$$

The equation is in standard form.

$$\left(xy+2x-y\right)d=0$$

Divide 0 by 2x+yx-y.

$$d=0$$

Solve for x (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply 2 by 1+y.

$$\left(2+2y\right)dx-\left(1+x\right)dy=0$$

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

$$\left(2d+2yd\right)x-\left(1+x\right)dy=0$$

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

$$2dx+2ydx-\left(1+x\right)dy=0$$

Use the distributive property to multiply 1+x by d.

$$2dx+2ydx-\left(d+xd\right)y=0$$

Use the distributive property to multiply d+xd by y.

$$2dx+2ydx-\left(dy+xdy\right)=0$$

To find the opposite of dy+xdy, find the opposite of each term.

$$2dx+2ydx-dy-xdy=0$$

Combine 2ydx and -xdy to get ydx.

$$2dx+ydx-dy=0$$

Add dy to both sides. Anything plus zero gives itself.

$$2dx+ydx=dy$$

Combine all terms containing x.

$$\left(2d+yd\right)x=dy$$

The equation is in standard form.

$$\left(dy+2d\right)x=dy$$

Divide both sides by 2d+yd.

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

Dividing by 2d+yd undoes the multiplication by 2d+yd.

$$x=\frac{dy}{dy+2d}$$

Divide dy by 2d+yd.

$$x=\frac{y}{y+2}$$

Solve for d

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

Steps for Solving Linear Equation

Use the distributive property to multiply 2 by 1+y.

$$\left(2+2y\right)dx-\left(1+x\right)dy=0$$

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

$$\left(2d+2yd\right)x-\left(1+x\right)dy=0$$

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

$$2dx+2ydx-\left(1+x\right)dy=0$$

Use the distributive property to multiply 1+x by d.

$$2dx+2ydx-\left(d+xd\right)y=0$$

Use the distributive property to multiply d+xd by y.

$$2dx+2ydx-\left(dy+xdy\right)=0$$

To find the opposite of dy+xdy, find the opposite of each term.

$$2dx+2ydx-dy-xdy=0$$

Combine 2ydx and -xdy to get ydx.

$$2dx+ydx-dy=0$$

Combine all terms containing d.

$$\left(2x+yx-y\right)d=0$$

The equation is in standard form.

$$\left(xy+2x-y\right)d=0$$

Divide 0 by 2x+yx-y.

$$d=0$$

Solve for x

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

Steps for Solving Linear Equation

Use the distributive property to multiply 2 by 1+y.

$$\left(2+2y\right)dx-\left(1+x\right)dy=0$$

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

$$\left(2d+2yd\right)x-\left(1+x\right)dy=0$$

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

$$2dx+2ydx-\left(1+x\right)dy=0$$

Use the distributive property to multiply 1+x by d.

$$2dx+2ydx-\left(d+xd\right)y=0$$

Use the distributive property to multiply d+xd by y.

$$2dx+2ydx-\left(dy+xdy\right)=0$$

To find the opposite of dy+xdy, find the opposite of each term.

$$2dx+2ydx-dy-xdy=0$$

Combine 2ydx and -xdy to get ydx.

$$2dx+ydx-dy=0$$

Add dy to both sides. Anything plus zero gives itself.

$$2dx+ydx=dy$$

Combine all terms containing x.

$$\left(2d+yd\right)x=dy$$

The equation is in standard form.

$$\left(dy+2d\right)x=dy$$

Divide both sides by 2d+yd.

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

Dividing by 2d+yd undoes the multiplication by 2d+yd.

$$x=\frac{dy}{dy+2d}$$

Divide dy by 2d+yd.

$$x=\frac{y}{y+2}$$