eqn1=x1'[t]==2*x1[t]+x2[t];
eqn2=x2'[t]==2*x1[t]-x2[t];
eqn3=x1[0]==x10;
eqn4=x2[0]==x20;

A={{2,1},{2,-1}};

Eigenvalues[A]

N[Eigenvalues[A]]

sol=DSolve[ {eqn1,eqn2,eqn3,eqn4},{x1,x2},t];

Simplify[x1[t]/.sol]

Simplify[x2[t]/.sol]

N[Simplify[x1[t]/.sol]]

N[Simplify[x2[t]/.sol]]

ParametricPlot[Evaluate[Join[{x1[t],x2[t]}/.NDSolve[{eqn1,eqn2,x1[0]==#,x2[0]==-10},{x1[t],x2[t]},{t,-2,2}][[1]]&/@{-17.8078,2.8078},
{x1[t],x2[t]}/.NDSolve[{eqn1,eqn2,x1[0]==-10,x2[0]==#},{x1[t],x2[t]},{t,-2,2}][[1]]&/@{Sequence@@Range[-10,10,1]},{x1[t],x2[t]}/.NDSolve[{eqn1,eqn2,x1[0]==#,x2[0]==10},{x1[t],x2[t]},{t,-2,2}][[1]]&/@{-2.8078,17.8078},
{x1[t],x2[t]}/.NDSolve[{eqn1,eqn2,x1[0]==10,x2[0]==#},{x1[t],x2[t]},{t,-2,2}][[1]]&/@{Sequence@@Range[-10,10,1]}
]],{t,-2,2},PlotRange->{{-10,10},{-10,10}},PlotStyle->Blue,DisplayFunction->Identity,AxesLabel->TraditionalForm/@{x1,x2}, LabelStyle->Directive[12],Background->White]