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

Nov 21, 2017

$y = B {e}^{x} - c$

#### Explanation:

We have:

$\frac{\mathrm{dy}}{\mathrm{dx}} = y + c$

This is a First Order Linear Differential Equation which we can rewrite as a separable equation and thus "separate the variables" to get:

$\frac{1}{y + c} \setminus \frac{\mathrm{dy}}{\mathrm{dx}} = 1$

$\therefore \int \setminus \frac{1}{y + c} \setminus \mathrm{dy} = \int \setminus \mathrm{dx}$

Which consists of standard integral function ; so we can integrate to get

$\ln | y + c | = x + A$

Taking exponentials we get:

$| y + c | = {e}^{x + A}$

And as the exponential is positive for all values; we must have:

$y + c = {e}^{x} {e}^{A}$

Which we can write as:

$y = B {e}^{x} - c$