ClearAll

(*V=ListVectorPlot[Table[{y,-x+y},{x,-5,5,0.1},{y,-5,5,0.1}]]*)

DSolve[{x'[t]==y[t],y'[t]==-x[t]+y[t],x[0]==1,y[0]==0},{x[t],y[t]},t](*x1*)

x1=ParametricPlot[{(1/3) (E^(t/2)) (3 Cos[(((Sqrt[3]) t)/2)]-(Sqrt[3]) Sin[(((Sqrt[3]) t)/2)]),-((2 (E^(t/2)) Sin[(((Sqrt[3]) t)/2)])/(Sqrt[3])) }, {t, -10,10}, PlotStyle->Red];

DSolve[{x'[t]==y[t],y'[t]==-x[t]+y[t],x[0]==0,y[0]==-1},{x[t],y[t]},t](*x2*)

x2=ParametricPlot[{-((2 (E^(t/2)) Sin[(((Sqrt[3]) t)/2)])/(Sqrt[3])),-(1/3) (E^(t/2)) (3 Cos[(((Sqrt[3]) t)/2)]+(Sqrt[3]) Sin[(((Sqrt[3]) t)/2)]) }, {t, -10,10}, PlotStyle->Red];

DSolve[{x'[t]==y[t],y'[t]==-x[t]+y[t],x[0]==-1,y[0]==0},{x[t],y[t]},t](*x3*)

x3=ParametricPlot[{-(1/3) (E^(t/2)) (3 Cos[(((Sqrt[3]) t)/2)]-(Sqrt[3]) Sin[(((Sqrt[3]) t)/2)]),((2 (E^(t/2)) Sin[(((Sqrt[3]) t)/2)])/(Sqrt[3])) }, {t, -10,10}, PlotStyle->Red];

DSolve[{x'[t]==y[t],y'[t]==-x[t]+y[t],x[0]==0,y[0]==1},{x[t],y[t]},t](*x4*)

x4=ParametricPlot[{((2 (E^(t/2)) Sin[(((Sqrt[3]) t)/2)])/(Sqrt[3])),(1/3) (E^(t/2)) (3 Cos[(((Sqrt[3]) t)/2)]+(Sqrt[3]) Sin[(((Sqrt[3]) t)/2)]) }, {t, -10,10}, PlotStyle->Red];

v2=VectorPlot[{y,-x+y}, {x, -10,10}, {y, -10,10}];

Show[v2,x1,x2,x3,x4]