What is a solution to the differential equation (dy/dx) +5y = 9?

Nov 21, 2016

$y = \frac{9}{5} + B {e}^{- 5 x}$

Explanation:

This is a First Order Separable Differential Equation.

We can isolate the variables as follows;

$\frac{\mathrm{dy}}{\mathrm{dx}} + 5 y = 9$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = 9 - 5 y$

Separating the variable gives us:

$\therefore \int \frac{1}{9 - 5 y} \mathrm{dy} = \int \mathrm{dx}$
$\therefore - \int \frac{1}{5 y - 9} \mathrm{dy} = \int \mathrm{dx}$

Integrating gives us:

$- \frac{1}{5} \ln | 5 y - 9 | = x + C$
$\therefore \ln | 5 y - 9 | = - 5 x - 5 C$
$\therefore 5 y - 9 = {e}^{- 5 x - 5 C}$
$\therefore 5 y - 9 = {e}^{- 5 x} {e}^{- 5 C}$
$\therefore 5 y - 9 = A {e}^{- 5 x}$
$\therefore 5 y = 9 + \frac{A}{5} {e}^{- 5 x}$
$\therefore y = \frac{9}{5} + B {e}^{- 5 x}$

Verification of Solution:

$y = \frac{9}{5} + B {e}^{- 5 x}$
$y = - 5 B {e}^{- 5 x}$

So, $\frac{\mathrm{dy}}{\mathrm{dx}} + 5 y = - 5 B {e}^{- 5 x} + 5 \left\{\frac{9}{5} + B {e}^{- 5 x}\right\}$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} + 5 y = - 5 B {e}^{- 5 x} + 9 + 5 B {e}^{- 5 x}$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} + 5 y = 9$ QED