xpartic[t_] := Cos[10 t]
w0=Sqrt[91.]/2.;
beta=1.5;
delta=ArcTan[beta/w0];
xhomog[t_] := Exp[-beta t] * Cos[w0*t - delta]
Plot[{xpartic[t],xhomog[t]},{t,0,16Pi/10}]

Plot[xpartic[t]+xhomog[t],{t,0,16 Pi/10}]

xpartic2[t_] := Cos[8 t]
w2=32.;
beta2=3;
delta2=ArcTan[beta/w0];
xhomog2[t_] := Exp[-beta2 t] * Cos[w2*t - delta2]
Plot[{xpartic2[t],xhomog2[t]},{t,0,8 Pi/10}]

Plot[xpartic2[t]+xhomog2[t],{t,0,8 Pi/10}]