vectorField[x_,y_] := {y,x}

vectorFieldPlot[vf_] := VectorPlot[vectorField[x,y],{x,-4,4},{y,-4,4},VectorScale->{.035,.9,None},VectorPoints->30,PlotRange->Automatic]

flowLine[t_] := {Cosh[t],Sinh[t]}

flowLinePlot[fl_] := ParametricPlot[fl,{t,0,5}, PlotRange->4,PlotStyle->{Thick,Red}]

Show[{vectorFieldPlot[vectorField],flowLinePlot[flowLine[t]]}]