Manipulate[Plot[{k1 Exp[-t]+k2 Exp[-4t]+(3Sin[t]-5Cos[t])/34,Sin[t+ArcTan[-5/3]]/Sqrt[34]},{t,0,20},PlotRange->{-.5,.5}],{k1,-4,4},{k2,-4,4}]

y[t_]:=k1 Exp[-t]+k2 Exp[-4t]+(3Sin[t]-5Cos[t])/34

v[t_]:=y'[t]

v[t]

ParametricPlot[{y[t],v[t]}/.{k1->2,k2->-1},{t,0,20}]