# How to you find the general solution of dy/dx=x/(1+x^2)?

May 10, 2017

Multiply both sides by $\mathrm{dx}$ and then integrate.

#### Explanation:

Given: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{x}{1 + {x}^{2}}$

Multiply both sides by $\mathrm{dx}$

$\mathrm{dy} = \frac{x}{1 + {x}^{2}} \mathrm{dx}$

Integrate:

$\int \mathrm{dy} = \int \frac{x}{1 + {x}^{2}} \mathrm{dx}$

$\int \mathrm{dy} = \frac{1}{2} \int \frac{2 x}{1 + {x}^{2}} \mathrm{dx}$

$\int \mathrm{dy} = \frac{1}{2} \frac{\mathrm{du}}{u}$

$y = \frac{1}{2} \ln \left(u\right) + C$

$y = \frac{1}{2} \ln \left({x}^{2} + 1\right) + C$

$y = \ln \left(\sqrt{{x}^{2} + 1}\right) + C$