# How to you find the general solution of (dr)/(ds)=0.05r?

May 24, 2017

$r = A {e}^{0.05 s}$

#### Explanation:

We have:

$\frac{\mathrm{dr}}{\mathrm{ds}} = 0.05 r$

Which is a First Order linear separable DE. We can simply separate the variables to get

$\int \setminus \frac{1}{r} \setminus \mathrm{dr} = \int \setminus 0.05 \setminus \mathrm{ds}$

Then integrating gives:

$\setminus \setminus \ln | r | = 0.05 s + C$
$\therefore | r | = {e}^{0.05 s + C}$
$\therefore | r | = {e}^{0.05 s} \cdot {e}^{C}$

And as ${e}^{x} > 0 \forall x \in \mathbb{R}$, and putting $A = {e}^{C}$ we can write the solution as:

$r = A {e}^{0.05 s}$