Solve dz/z^2-z=dx/x

$$\frac { d z } { z ^ { 2 } - z } = \frac { d x } { x }$$

Solve for d

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

Steps for Solving Linear Equation

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

$$xdz=z\left(z-1\right)dx$$

Use the distributive property to multiply z by z-1.

$$xdz=\left(z^{2}-z\right)dx$$

Use the distributive property to multiply z^{2}-z by d.

$$xdz=\left(z^{2}d-zd\right)x$$

Use the distributive property to multiply z^{2}d-zd by x.

$$xdz=z^{2}dx-zdx$$

Subtract z^{2}dx from both sides.

$$xdz-z^{2}dx=-zdx$$

Add zdx to both sides.

$$xdz-z^{2}dx+zdx=0$$

Combine xdz and zdx to get 2xdz.

$$2xdz-z^{2}dx=0$$

Combine all terms containing d.

$$\left(2xz-z^{2}x\right)d=0$$

The equation is in standard form.

$$\left(2xz-xz^{2}\right)d=0$$

Divide 0 by 2xz-z^{2}x.

$$d=0$$

Solve for x

$$x\neq 0$$

$$\left(d=0\text{ and }z\neq 1\text{ and }z\neq 0\right)\text{ or }z=2$$

Steps for Solving Linear Equation

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by xz\left(z-1\right), the least common multiple of z^{2}-z,x.

$$xdz=z\left(z-1\right)dx$$

Use the distributive property to multiply z by z-1.

$$xdz=\left(z^{2}-z\right)dx$$

Use the distributive property to multiply z^{2}-z by d.

$$xdz=\left(z^{2}d-zd\right)x$$

Use the distributive property to multiply z^{2}d-zd by x.

$$xdz=z^{2}dx-zdx$$

Subtract z^{2}dx from both sides.

$$xdz-z^{2}dx=-zdx$$

Add zdx to both sides.

$$xdz-z^{2}dx+zdx=0$$

Combine xdz and zdx to get 2xdz.

$$2xdz-z^{2}dx=0$$

Combine all terms containing x.

$$\left(2dz-z^{2}d\right)x=0$$

The equation is in standard form.

$$\left(2dz-dz^{2}\right)x=0$$

Divide 0 by 2dz-z^{2}d.

$$x=0$$

Variable x cannot be equal to 0.

$$x\in \emptyset$$