Answer
$$y = - \frac{1}{2}{e^{ - 2t}} + \frac{{13}}{2}$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dt}} = {e^{ - 2t}} \cr
& {\text{Separate the variables}} \cr
& dy = {e^{ - 2t}}dt \cr
& {\text{Integrate both sides with respecto to }}x \cr
& \int {dy} = \int {{e^{ - 2t}}} dt \cr
& y = - \frac{1}{2}{e^{ - 2t}} + C \cr
& {\text{Apply the initial condition }}y\left( 0 \right) = 6 \cr
& 6 = - \frac{1}{2}{e^{ - 2\left( 0 \right)}} + C \cr
& {\text{Simplifying}} \cr
& 6 = - \frac{1}{2} + C \cr
& C = \frac{{13}}{2} \cr
& {\text{Substitute }}C{\text{ into }}y = - \frac{1}{2}{e^{ - 2t}} + C \cr
& y = - \frac{1}{2}{e^{ - 2t}} + \frac{{13}}{2} \cr} $$
