Solve y^2/y+1dy=dx/x^2

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

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq -1\text{ and }x\neq 0\\d\in \mathrm{R}\text{, }&x=\frac{y+1}{y^{3}}\text{ and }y\neq 0\text{ and }y\neq -1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(y+1\right)x^{2}, the least common multiple of y+1,x^{2}.

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{2}y^{3}d=\left(y+1\right)dx$$

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

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

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

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

Subtract ydx from both sides.

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

Subtract dx from both sides.

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

Reorder the terms.

$$dx^{2}y^{3}-dxy-dx=0$$

Combine all terms containing d.

$$\left(x^{2}y^{3}-xy-x\right)d=0$$

Divide 0 by x^{2}y^{3}-xy-x.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{y+1}{y^{3}}\text{, }&y\neq -1\text{ and }y\neq 0\\x\neq 0\text{, }&d=0\text{ and }y\neq -1\end{matrix}\right.$$