Needs["VectorFieldPlots`"]

a=-1;b=10;c=-1;d=-1;

vektorikentta=VectorFieldPlot[{a x+b y,c x+d y},{x,-6,6},{y,-6,6}] ;
dy[{x0_,y0_}]:=NDSolve[{x'[t]==a x[t]+b y[t],y'[t]==c x[t]+d y[t],x[0]==x0,y[0]==y0},{x[t],y[t]},{t,-5,10}];

Show[vektorikentta,AxesLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},Axes->True]

alkuarvo={6,1};
ratk=dy[alkuarvo];

ratkaisu=ParametricPlot[{x[t],y[t]}/.ratk,{t,0,10}, PlotRange->{{-6,6},{-6,6}}, PlotPoints->100,AxesLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},PlotStyle->Thickness[0.005]]

Show[{vektorikentta,ratkaisu,Graphics[{Red,PointSize[.02],Point[alkuarvo]}]},AxesLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},Axes->True, PlotRange->{{-6,6},{-6,6}}]