# How to you find the general solution of dy/dx=5e^(-x/2)?

Nov 8, 2016

THe general solution is $y = - 10 {e}^{- \frac{x}{2}} + C$

#### Explanation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = 5 {e}^{- \frac{x}{2}}$
$\therefore \mathrm{dy} = 5 {e}^{- \frac{x}{2}} \mathrm{dx}$
Integrating both sides
$\int \mathrm{dy} = 5 \int {e}^{- \frac{x}{2}} \mathrm{dx}$
We use the substitution, $u = - \frac{x}{2}$ $\implies$$- 2 \mathrm{du} = \mathrm{dx}$
$y = 5 \int - 2 {e}^{u} \mathrm{du} = - 10 {e}^{u} = - 10 {e}^{- \frac{x}{2}} + C$