# How to you find the general solution of (2+x)y'=3y?

Feb 16, 2017

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

#### Explanation:

We have;

$\left(2 + x\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 3 y$

This is a First Order separable Differential equation, and we can rearrange as follows:

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

And we can "separate the variables" to get:

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

This is straightforward to integrate:

$\setminus \ln y = 3 \ln \left(x + 2\right) + \ln A$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \ln {\left(x + 2\right)}^{3} + \ln A$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \ln A {\left(x + 2\right)}^{3}$
$\therefore y = A {\left(x + 2\right)}^{3}$