Answer
$$y\left( t \right) = - 2{e^{ - 4t}} + t + 7$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dt}} = 8{e^{ - 4t}} + 1 \cr
& {\text{separate the variables}} \cr
& dy = \left( {8{e^{ - 4t}} + 1} \right)dt \cr
& {\text{integrate both sides }} \cr
& \int {dy} = \int {\left( {8{e^{ - 4t}} + 1} \right)dt} \cr
& y\left( t \right) = \frac{8}{{ - 4}}{e^{ - 4t}} + t + C \cr
& y\left( t \right) = - 2{e^{ - 4t}} + t + C \cr
& {\text{the initial condition }}y\left( 0 \right) = 5{\text{ implies that}} \cr
& 5 = - 2{e^{ - 4\left( 0 \right)}} + 0 + C \cr
& C = 7 \cr
& {\text{so the solution of the initial value problem is}} \cr
& y\left( t \right) = - 2{e^{ - 4t}} + t + 7 \cr} $$
