# What is the general solution of the differential equation  dy/dx + 2y = 0?

Jul 22, 2017

$y = C {e}^{- 2 x}$

#### Explanation:

We have:

$\frac{\mathrm{dy}}{\mathrm{dx}} + 2 y = 0$

We can just rearrange as follows:

$\frac{\mathrm{dy}}{\mathrm{dx}} = - 2 y \implies \frac{1}{y} \setminus \frac{\mathrm{dy}}{\mathrm{dx}} = - 2$

This is a first Order Separable Differential Equation and "separate the variables" to get

$\int \setminus \frac{1}{y} \setminus \mathrm{dy} = \int \setminus - 2 \setminus \mathrm{dx}$

And integrating gives us:

$\ln | y | = - 2 x + A$

$\therefore | y | = {e}^{- 2 x + A}$

Note that as ${e}^{\alpha} > 0 \forall \alpha \in \mathbb{R}$, we can write

$y = {e}^{- 2 x + A}$
$\setminus \setminus = {e}^{- 2 x} {e}^{A}$
$\setminus \setminus = C {e}^{- 2 x}$