Needs["VectorFieldPlots`"]

k:=-0.5;
VectorFieldPlot[{x2,-k*x1},{x1,-3,3},{x2,-3,3},Axes->True,PlotPoints->9]

ParametricPlot[{Sin[t],Cos[t]},{t,0,2*Pi}]

k:=-0.5;
VectorFieldPlot[{x2,-k*Sin[x1]},{x1,-2*Pi,2*Pi,Pi/4},{x2,-2,2,0.5},Axes->True,Ticks->{{-2*Pi,-Pi,0,Pi,2*Pi},Automatic}]

k:=2;
m:=1;
VectorFieldPlot[{x2,-k*x1-m*x2},{x1,-3,3},{x2,-3,3},Axes->True,PlotPoints->9]
m:=2;
VectorFieldPlot[{x2,-k*x1-m*x2},{x1,-3,3},{x2,-3,3},Axes->True,PlotPoints->9]

sol=DSolve[x'[t]==Cos[x[t]]^2,x[t],t]
Plot[{x[t]/.sol/.C[1]->0,x[t]/.sol/.C[1]->1,x[t]/.sol/.C[1]->2},{t,-2*Pi,2*Pi}]

sol1=DSolve[{x'[t]==Sin[x[t]],x[1]==Pi/2},x[t],t]

sol11=NDSolve[{x'[t]==Sin[x[t]],x[1]==Pi/2},x,{t,0,2*Pi}]
Plot[{Pi,x[t]/.sol11},{t,0,2*Pi}]

sol=DSolve[x'[t]==Sin[E^(x[t])],x[t],t]
Plot[{x[t]/.sol/.C[1]->0,x[t]/.sol/.C[1]->1,x[t]/.sol/.C[1]->2},{t,-2*Pi,2*Pi}]

sol11=NDSolve[{x'[t]==Sin[E^(x[t])],x[1]==0},x,{t,0,2*Pi}]
Plot[{Pi,x[t]/.sol11},{t,0,2*Pi}]