Syntax is incorrect.Use: "k*x[t]" not "kx[t]" !
Try:
Clear["Global'*"]
DSolve[x''[t] + b*x'[t] + k*x[t] == 0, x[t], t]
res1 = DSolve[{x''[t] + b*x'[t] + k*x[t] == 0, x[0] == 1, x'[0] == 0},
x[t], t]
res1plot[b0_] :=
Plot[x[t] /. res1 /. {b -> b0, k -> 1}, {t, 0, 20},
PlotRange -> {{0, 20}, {-1, 1}}, PlotStyle -> Thick, PlotLabel -> b0]
res1plot[0]
Clear["Global'*"]
DSolve[{x''[t] + b*x'[t] + k*x[t] == Sin[\[Omega]t]}, x[t], t]
res3 = DSolve[{x''[t] + b*x'[t] + k*x[t] == Sin[\[Omega]*t],
x[0] == 1, x'[0] == 0}, x[t], t];
res3f[t0_, omin_] =
x[t] /. res3 /. {b -> 0.1, k -> 1, \[Omega] -> omin} /. t -> t0;
res3plot[omega_] :=
Plot[res3f[t, omega], {t, 0, 200}, PlotRange -> {{0, 200}, {-5, 5}},
PlotStyle -> Thick, PlotLabel -> omega]
res3plot[.2]