First Order Linear D.E.'s with Constant Coefficients

y' = ry + k

assuming $r \neq 0$, and $y \neq - \frac{k}{r}$, we can write

$$\frac{y'}{y + \frac{k}{r}} = r$$

now integrate...

$$\ln{\vert y + \frac{k}{r}\vert} = rt + C$$

and take the exponential of both sides...

$$y + \frac{k}{r} = \pm e^Ce^{rt}$$

and finally, solve for y.

$$y = -\frac{k}{r} + ce^{rt}$$