Answer
$$I = \frac{{{E_0}}}{R} + C{e^{ - \frac{R}{L}t}}{\text{ }}$$
Work Step by Step
$$\eqalign{
& {\text{We have the differential equation }} \cr
& L\frac{{dI}}{{dt}} + RI = E \cr
& \frac{{dI}}{{dt}} + \frac{R}{L}I = \frac{E}{L} \cr
& {\text{The differential equation has the form }}\frac{{dI}}{{dt}} + P\left( t \right)I = Q\left( t \right) \cr
& {\text{With }}P\left( t \right) = \frac{R}{L},{\text{ }}Q\left( t \right) = \frac{E}{L} \cr
& {\text{Find the integrating factor }}I\left( x \right) = {e^{\int {Q\left( t \right)} dt}} \cr
& I\left( t \right) = {e^{\int {\frac{R}{L}} dt}} = {e^{\frac{R}{L}t}} \cr
& {\text{Multiply the differential equation by the integrating factor}} \cr
& {e^{\frac{R}{L}t}}\left( {\frac{{dI}}{{dt}} + \frac{R}{L}I = \frac{E}{L}} \right) = \frac{E}{L}{e^{\frac{R}{L}t}} \cr
& {e^{\frac{R}{L}t}}\frac{{dI}}{{dt}} + \frac{R}{L}{e^{\frac{R}{L}t}}I = \frac{E}{L}{e^{\frac{R}{L}t}} \cr
& {\text{Write the left side in the form }}\frac{d}{{dt}}\left[ {I\left( t \right)P} \right] \cr
& \frac{d}{{dt}}\left[ {{e^{\frac{R}{L}t}}I} \right] = \frac{E}{L}{e^{\frac{R}{L}t}} \cr
& d\left[ {{e^{\frac{R}{L}t}}I} \right] = \frac{E}{L}{e^{\frac{R}{L}t}}dt \cr
& {\text{Integrate both sides}} \cr
& {e^{\frac{R}{L}t}}I = \int {\frac{E}{L}{e^{\frac{R}{L}t}}} dt \cr
& {e^{\frac{R}{L}t}}I = \frac{{LE}}{{LR}}{e^{\frac{R}{L}t}} + C \cr
& {e^{\frac{R}{L}t}}I = \frac{E}{R}{e^{\frac{R}{L}t}} + C \cr
& {\text{Solve for }}I \cr
& I = \frac{E}{R} + C{e^{ - \frac{R}{L}t}}{\text{ }} \cr
& {\text{Let }}E = {E_0} \cr
& I = \frac{{{E_0}}}{R} + C{e^{ - \frac{R}{L}t}} \cr} $$
