vplot = VectorPlot[{-x1+2*x2,-x1-3*x2},{x1,-3,3},{x2,-3,3},VectorScale->{0.04,1,None},Axes->True]

u[t_] = {Exp[-2*t]*(-Cos[t] + Sin[t]),Exp[-2*t]*Cos[t]}

solplot1 = ParametricPlot[u[t],{t,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->Green];

v[t_] = {Exp[-2*t]*(-Cos[t] - Sin[t]),Exp[-2*t]*Sin[t]}

solplot2 = ParametricPlot[v[t],{t,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->Purple];

Show[vplot,solplot1,solplot2]

Manipulate[Show[
vplot,
solplot1,
solplot2,
ParametricPlot[c1*u[t] + c2*v[t],{t,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->Red]],{{c1,0},-5,5,Appearance->"Labeled"},{{c2,0},-5,5,Appearance->"Labeled"}]