# What is a solution to the differential equation dy/dx = 1 - 0.2y?

Jul 2, 2016

$y = \alpha {e}^{- \frac{1}{5} x} + 5$

#### Explanation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = 1 - 0.2 y$

$= \frac{1}{5} \left(5 - y\right)$

$\int \frac{1}{5 - y} \mathrm{dy} = \frac{1}{5} \int \mathrm{dx}$

$\int \frac{1}{y - 5} \mathrm{dy} = - \frac{1}{5} \int \mathrm{dx}$

$\ln \left(y - 5\right) = - \frac{1}{5} x + C$

$y - 5 = \exp \left(- \frac{1}{5} x + C\right)$

$= \alpha {e}^{- \frac{1}{5} x}$ [where $\alpha = {e}^{C}$]

$y = \alpha {e}^{- \frac{1}{5} x} + 5$