Clear[θ, α, β, Flowline1, 
      Flowline2]
{EV1[θ_], 
        EV2[θ_]} = {{θ, 
          2}, {2, θ}}
{α, β, Uinitial} = { .75, 
        1, {1, 1}}
Flowline1[θ_] = (∫_0^θ) 
              EV1[θ] \[DifferentialD]θ  + 
        Uinitial
Flowline2[θ1_, θ2_] = (∫_0^θ)2 
              EV2[θ] \[DifferentialD]θ + 
        Flowline1[θ1]
Show[
      ParametricPlot[
        Flowline1[θ], {θ, 
          0, α}], 
ParametricPlot[
        Flowline2[α, θ], {θ, 
          0, β}]]