# What is a solution to the differential equation dy/dx=4-6y?

Jul 9, 2016

$y = \frac{2 - C {e}^{- 6 x}}{3}$

#### Explanation:

this is separable

$\frac{\mathrm{dy}}{\mathrm{dx}} = 4 - 6 y$

$\frac{1}{4 - 6 y} \frac{\mathrm{dy}}{\mathrm{dx}} = 1$

we integrate both sides

$\int \frac{1}{4 - 6 y} \frac{\mathrm{dy}}{\mathrm{dx}} \setminus \mathrm{dx} = \int \setminus \mathrm{dx}$

or

$\int \frac{1}{4 - 6 y} \mathrm{dy} = \int \setminus \mathrm{dx}$

$- \frac{1}{6} \ln \left(4 - 6 y\right) = x + C$

please note that I am using C as a generic constant here so it's value changes through the process

$\ln \left(4 - 6 y\right) = C - 6 x$

$4 - 6 y = {e}^{C - 6 x} = {e}^{C} {e}^{- 6 x} = C {e}^{- 6 x}$

$y = \frac{2 - C {e}^{- 6 x}}{3}$

please note that I am using C as a generic constant here so it's value changes through the process