ev1=Graphics[{Arrowheads[Large],Arrow[{{0,0},{1,5}}]}];
ev2=Graphics[{Arrowheads[Large],Arrow[{{1,1},{0,0}}]}];

Show[ev1,ev2]

soln[c_,d_]:=NDSolve[{dx'[dt]==-2 dx[dt]+dy[dt],  dy'[dt]==-5 dx[dt]+4dy[dt]  ,dx[0]==c,dy[0]==d},{dx,dy},{dt,-2,2}]

soln1=soln[2,2.3]

trace1=ParametricPlot[{dx[t],dy[t]}/.soln1,{t,-.8,.8}]

soln2=soln[2,1.7]

trace2=ParametricPlot[{dx[t],dy[t]}/.soln2,{t,-.8,.8}]

soln3=soln[-2,-1.7]

trace3=ParametricPlot[{dx[t],dy[t]}/.soln3,{t,-.8,.8}]

soln4=soln[-2,-2.3]

trace4=ParametricPlot[{dx[t],dy[t]}/.soln4,{t,-.8,.8}]

Show[trace1,ev1,ev2,trace2,trace3,trace4,PlotRange->All,AxesOrigin->{0,0}]

lsoln[c_,d_]:=NDSolve[{dx'[dt]== dx[dt]+2 dy[dt],  dy'[dt]==-5 dx[dt]-dy[dt]  ,dx[0]==c,dy[0]==d},{dx,dy},{dt,-2,2}]

lsoln1=lsoln[1,0]

ltrace1=ParametricPlot[{dx[t],dy[t]}/.lsoln1,{t,0,2}];

lsoln2=lsoln[1/2,0]

ltrace2=ParametricPlot[{dx[t],dy[t]}/.lsoln2,{t,0,1.5}];

direction=Graphics[{Arrowheads[Large],Arrow[{{1,0},{5/4,-5/4}}]}];

Show[ltrace1,ltrace2,direction,PlotRange->All]