Solución General

DSolve[(1/2) i'[t] + 10  i[t] == 12, i, t]

DSolve[(1/2) i'[t] + 10  i[t] == 12, i[t], t]

Solución Particular

Sol = 
      DSolve[{(1/2) i'[t] + 10  i[t] == 12, i[0] == 0}, 
        i[t], t]

Simplify[Sol]

Solución Transitoria

fig1 = 
        Plot[(6/5) - ((6 (E^(-20 t)))/5), {t, 0,  .33}, 
          AxesLabel -> TraditionalForm /@ {t[s], i [A]}, 
          TextStyle -> {FontFamily -> "Times", 
              FontSize -> 14}, PlotRange -> {- .05, 1.3}, 
          PlotStyle -> {Hue[0.8]}];

Solución Estacionaria

\(\(fig2 = 
        Plot[6\/5 - \(6\ E\^\(\(-20\)\ t\)\)\/5, {t,  .33,  .8}, AxesLabel -> TraditionalForm /@ {t[s], i\ [A]}, 
          TextStyle -> {FontFamily -> "Times", 
              FontSize -> 14}, PlotRange -> {\(- .05\), 1.3}, 
          PlotStyle -> {Hue[0.26]}];\)\)

Solución i (t)

Show[fig1, fig2]