Q[a_,b_]={{a,b},{-b,a}}; MatrixForm[Q[a,b]]

x[t]={x1[t],x2[t]};

gensol=DSolve[D[x[t],t]==Q[a,b].x[t],x[t],t]

Y[a_,b_,t_]=Simplify[{gensol[[1,1,2]],gensol[[1,2,2]]}]

Simplify[D[Y[a,b,t],t]-Q[a,b].Y[a,b,t]]

StreamPlot[Q[.2,1].{x,y},{x,-1,1},{y,-1,1}]

sol1[t_]=Flatten[Table[Flatten[{t,Y[.2,1,t]}]/.{C[1]->i,C[2]-> j},{i,-1,1,.5},{j,-1,1,.5}],1];

ParametricPlot3D[sol1[t],{t,0,2Pi}]

Animate[Show[ParametricPlot3D[sol1[t],{t,0,x},PlotRange->{{-.5,2Pi},{-4.5,4.5},{-4.5,4.5}}],ParametricPlot3D[{t,0,0},{t,0,2Pi},PlotStyle->{Red}]],{x,0,2Pi}]

Animate[Show[ParametricPlot3D[Flatten[{t,Y[.2,1,t]}]/.{C[1]->1,C[2]-> 1},{t,0,x},PlotRange->{{-.5,4Pi},{-20.5,20.5},{-20.5,20.5}}],ParametricPlot3D[{t,0,0},{t,0,4Pi},PlotStyle->{Red}]],{x,0,4Pi}]

StreamPlot[Q[-.2,1].{x,y},{x,-1,1},{y,-1,1}]

sol2[t_]=Flatten[Table[Flatten[{t,Y[-.2,1,t]}]/.{C[1]->i,C[2]-> j},{i,-1,1,.5},{j,-1,1,.5}],1];

ParametricPlot3D[sol2[t],{t,0,4Pi}]

Animate[Show[ParametricPlot3D[sol2[t],{t,0,x},PlotRange->{{-.5,6Pi},{-1.5,1.5},{-1.5,1.5}}],ParametricPlot3D[{t,0,0},{t,0,6Pi},PlotStyle->{Red}]],{x,0,6Pi}]

Animate[Show[ParametricPlot3D[Flatten[{t,Y[-.2,1,t]}]/.{C[1]->1,C[2]-> 1},{t,0,x},PlotRange->{{-.5,6Pi},{-1.5,1.5},{-1.5,1.5}}],ParametricPlot3D[{t,0,0},{t,0,6Pi},PlotStyle->{Red}]],{x,0,6Pi}]

StreamPlot[Q[0,1].{x,y},{x,-1,1},{y,-1,1}]

sol3[t_]=Flatten[Table[Flatten[{t,Y[0,1,t]}]/.{C[1]->i,C[2]-> j},{i,-1,1,.5},{j,-1,1,.5}],1];

ParametricPlot3D[sol3[t],{t,0,2Pi}]

Animate[Show[ParametricPlot3D[sol3[t],{t,0,x},PlotRange->{{-.5,2Pi},{-1.5,1.5},{-1.5,1.5}}],ParametricPlot3D[{t,0,0},{t,0,2Pi},PlotStyle->{Red}]],{x,0,2Pi}]

Animate[Show[ParametricPlot3D[Flatten[{t,Y[0,1,t]}]/.{C[1]->1,C[2]-> 1},{t,0,x},PlotRange->{{-.5,4Pi},{-1.5,1.5},{-1.5,1.5}}],ParametricPlot3D[{t,0,0},{t,0,4Pi},PlotStyle->{Red}]],{x,0,4Pi}]