# What is a solution to the differential equation dy/dx=(3y)/(2+x)?

Jul 19, 2016

$y = {e}^{3 C} {\left(2 + x\right)}^{3}$

$= C {\left(2 + x\right)}^{3}$

#### Explanation:

This differential equation is separable.

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 y}{2 + x}$

$\mathrm{dy} = \frac{3 y}{2 + x} \mathrm{dx}$

$\frac{1}{3 y} \mathrm{dy} = \frac{1}{2 + x} \mathrm{dx}$

$\int \frac{1}{3 y} \mathrm{dy} = \int \frac{1}{2 + x} \mathrm{dx}$

$\frac{1}{3} \int \frac{1}{y} \mathrm{dy} = \int \frac{1}{2 + x} \mathrm{dx}$

$\frac{1}{3} \ln y = \ln \left(2 + x\right) + C$

$\ln y = 3 \left[\ln \left(2 + x\right) + C\right]$

$y = {e}^{3 \left[\ln \left(2 + x\right) + C\right]}$

$y = {e}^{3 C} {\left(2 + x\right)}^{3}$

$y = \textcolor{red}{C} {\left(2 + x\right)}^{3}$

[where C is generic constant]