Answer
$$u = \frac{7}{2}{e^{2t}} + \frac{1}{2}$$
Work Step by Step
$$\eqalign{
& u'\left( t \right) = 4u - 2 \cr
& \frac{{du}}{{dt}} = 4u - 2 \cr
& {\text{separate}} \cr
& \frac{{du}}{{4u - 2}} = dt \cr
& {\text{integrate both sides}} \cr
& \int {\frac{{du}}{{4u - 2}}} = \int {dt} \cr
& \frac{1}{4}\ln \left| {4u - 2} \right| = t + C \cr
& {\text{the initial condition }}u\left( 0 \right) = 4{\text{ implies that}} \cr
& \frac{1}{4}\ln \left| {4\left( 4 \right) - 2} \right| = 0 + C \cr
& C = \frac{1}{4}\ln 14 \cr
& \frac{1}{4}\ln \left| {4u - 2} \right| = t + \frac{1}{4}\ln 14 \cr
& {\text{solve for }}u \cr
& \ln \left| {4u - 2} \right| = 2t + \ln 14 \cr
& 4u - 2 = {e^{\ln 14}}{e^{2t}} \cr
& 4u - 2 = 14{e^{2t}} \cr
& u = \frac{7}{2}{e^{2t}} + \frac{1}{2} \cr} $$
