ParametricPlot[Evaluate[Table[{x[t],v[t]}/.DSolve[{x'[t]==v[t],v'[t]==-0.5x[t],x[0]==c,v[0]==0},{x[t],v[t]},t][[1]],{c,3}]],{t,0,10},PlotRange->All,PlotStyle->Directive[Red,AbsoluteThickness[3]],AxesLabel->{x,v},LabelStyle->{FontSize->20,Directive[Bold]}]

ParametricPlot[Evaluate[{x[t],v[t]}/.DSolve[{x'[t]==v[t],v'[t]==-0.5x[t],x[0]==1,v[0]==0},{x[t],v[t]},t][[1]]],{t,0,10},PlotRange->All,AxesLabel->{x,v},LabelStyle->{FontSize->30,Directive[Bold]},PlotStyle->{Red,AbsoluteThickness[3]}]

ParametricPlot[Evaluate[{x[t],v[t]}/.DSolve[{x'[t]==v[t],v'[t]==-0.5x[t]-0.02v[t],x[0]==1,v[0]==0},{x[t],v[t]},t][[1]]],{t,0,70},PlotRange->All,AxesLabel->{x,v},LabelStyle->{FontSize->20,Directive[Bold]},PlotStyle->{Red,AbsoluteThickness[3]}]

Plot[{τ=(Sqrt[4Δ]),v=-(Sqrt[4Δ])},{Δ,0,5},AxesLabel->{Δ,"τ"},Filling->Axis,FillingStyle->{LightOrange, LightYellow}, AspectRatio->Automatic,LabelStyle->{FontSize->20,Directive[Bold]},PlotStyle->{Directive[Red,AbsoluteThickness[5]]}]

Manipulate[ParametricPlot[Table[Table[{d Exp[a t],c Exp[-t]},{c,-3,3}],{d,-3,3}],{t,-10,10},PlotRange->12,AxesLabel->{x,y},LabelStyle->{FontSize->20,Directive[Bold]},PlotStyle->{Red,AbsoluteThickness[3]}],{a,-2,2}]