Chapter 6 - Differential Equations - 6.2 Exercises - Page 412: 4

$y=6+Ce^{-x} $

$\frac{dy}{dx}= 6-y$, Multiply both sides by dx then integrate $ \int \frac{1}{6-y}dy= \int dx$ Using u-substitution, let $ u=6-y$, and $-du=dy$ $ -\int \frac{1}{u}du = x +C$ $-\ln{|6-y|} = x+C$ $\ln{6-y}= -x-C$ Exponentiate to get $ |6-y|= e^{-x-C} $, Let $e^{-C} =C$ $|6-y|= Ce^{-x} $ $6-y= ^+_-Ce^{-x}$ $y= 6 ^+_-Ce^{-x}$, Let $ ^+_-C= C$ $y=6+Ce^{-x} $