Solve 1/ydy=x(x^2+1)^-1dx

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

Solve for d

$$d=0$$

$$y\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by y.

$$1dy=x\left(x^{2}+1\right)^{-1}dxy$$

Multiply x and x to get x^{2}.

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

Subtract x^{2}\left(x^{2}+1\right)^{-1}dy from both sides.

$$1dy-x^{2}\left(x^{2}+1\right)^{-1}dy=0$$

Reorder the terms.

$$dy-\frac{1}{x^{2}+1}dyx^{2}=0$$

Multiply both sides of the equation by x^{2}+1.

$$dy\left(x^{2}+1\right)-\frac{1}{x^{2}+1}dyx^{2}\left(x^{2}+1\right)=0$$

Use the distributive property to multiply dy by x^{2}+1.

$$dyx^{2}+dy-\frac{1}{x^{2}+1}dyx^{2}\left(x^{2}+1\right)=0$$

Express \frac{1}{x^{2}+1}d as a single fraction.

$$dyx^{2}+dy-\frac{d}{x^{2}+1}yx^{2}\left(x^{2}+1\right)=0$$

Express \frac{d}{x^{2}+1}y as a single fraction.

$$dyx^{2}+dy-\frac{dy}{x^{2}+1}x^{2}\left(x^{2}+1\right)=0$$

Express \frac{dy}{x^{2}+1}x^{2} as a single fraction.

$$dyx^{2}+dy-\frac{dyx^{2}}{x^{2}+1}\left(x^{2}+1\right)=0$$

Express \frac{dyx^{2}}{x^{2}+1}\left(x^{2}+1\right) as a single fraction.

$$dyx^{2}+dy-\frac{dyx^{2}\left(x^{2}+1\right)}{x^{2}+1}=0$$

Cancel out x^{2}+1 in both numerator and denominator.

$$dyx^{2}+dy-dyx^{2}=0$$

Combine dyx^{2} and -dyx^{2} to get 0.

$$dy=0$$

The equation is in standard form.

$$yd=0$$

Divide 0 by y.

$$d=0$$

Solve for x

$$x\in \mathrm{R}$$

$$d=0\text{ and }y\neq 0$$