Solve 1/y+1dy=(2x+1)dx

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Multiply both sides of the equation by y+1.

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

Use the distributive property to multiply 2x+1 by d.

$$1dy=\left(2xd+d\right)x\left(y+1\right)$$

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

$$1dy=\left(2dx^{2}+dx\right)\left(y+1\right)$$

Use the distributive property to multiply 2dx^{2}+dx by y+1.

$$1dy=2dx^{2}y+2dx^{2}+dxy+dx$$

Subtract 2dx^{2}y from both sides.

$$1dy-2dx^{2}y=2dx^{2}+dxy+dx$$

Subtract 2dx^{2} from both sides.

$$1dy-2dx^{2}y-2dx^{2}=dxy+dx$$

Subtract dxy from both sides.

$$1dy-2dx^{2}y-2dx^{2}-dxy=dx$$

Subtract dx from both sides.

$$1dy-2dx^{2}y-2dx^{2}-dxy-dx=0$$

Reorder the terms.

$$dy-2dyx^{2}-2dx^{2}-dxy-dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

Multiply both sides of the equation by y+1.

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

Use the distributive property to multiply 2x+1 by d.

$$1dy=\left(2xd+d\right)x\left(y+1\right)$$

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

$$1dy=\left(2dx^{2}+dx\right)\left(y+1\right)$$

Use the distributive property to multiply 2dx^{2}+dx by y+1.

$$1dy=2dx^{2}y+2dx^{2}+dxy+dx$$

Subtract 2dx^{2}y from both sides.

$$1dy-2dx^{2}y=2dx^{2}+dxy+dx$$

Subtract 2dx^{2} from both sides.

$$1dy-2dx^{2}y-2dx^{2}=dxy+dx$$

Subtract dxy from both sides.

$$1dy-2dx^{2}y-2dx^{2}-dxy=dx$$

Subtract dx from both sides.

$$1dy-2dx^{2}y-2dx^{2}-dxy-dx=0$$

Reorder the terms.

$$dy-2dyx^{2}-2dx^{2}-dxy-dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(y+1\right)\left(9y+1\right)}-y-1}{4\left(y+1\right)}\text{; }x=-\frac{\sqrt{\left(y+1\right)\left(9y+1\right)}+y+1}{4\left(y+1\right)}\text{, }&y\neq -1\\x\in \mathrm{C}\text{, }&d=0\text{ and }y\neq -1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(y+1\right)\left(9y+1\right)}-y-1}{4\left(y+1\right)}\text{; }x=-\frac{\sqrt{\left(y+1\right)\left(9y+1\right)}+y+1}{4\left(y+1\right)}\text{, }&y\geq -\frac{1}{9}\text{ or }y<-1\\x\in \mathrm{R}\text{, }&d=0\text{ and }y\neq -1\end{matrix}\right.$$