Play 1

eq1= x''[t]+ 2 γ x'[t]+ ω0^2 x[t] ==0

dsol1=DSolve[{eq1 /.γ->0,x'[0]==v0,x[0]==x0},x[t],t][[1]] //Simplify

Play 2

eq1= x''[t]+ 2 γ x'[t]+ ω0^2 x[t] ==0

dsol1=DSolve[{eq1 ,x'[0]==v0,x[0]==x0},x[t],t][[1]]  /.{(Sqrt[(γ^2)-((ω0)^2)])->I ω1} //FullSimplify

Play 3

eq1= x''[t]+ 2 γ x'[t]+ ω0^2 x[t] ==q0 Exp[I ωd t]

dsol1=DSolve[{eq1 /.γ->0,x'[0]==v0,x[0]==x0},x[t],t][[1]] //FullSimplify //Apart

Play 4

eq1= x''[t]+ 2 γ x'[t]+ ω0^2 x[t] ==q0 Exp[I ωd t]

dsol1=DSolve[{eq1 ,x'[0]==v0,x[0]==x0},x[t],t][[1]] //FullSimplify

(x[t] /.dsol1 )//Variables

{q0,v0,x0,γ,ω0,ωd}{q0,v0,x0,γ,ω0,ωd}

ttt={q0,v0,x0,γ,ω0,ωd} ->Hold[RandomReal[]] //Thread  //Release

Plot[ Re[x[t]] /.dsol1 //.ttt ,{t,0,50}]

Re[x[t]] /.dsol1

Manipulate:

Manipulate[
Plot[ (1/2) Re[((E^(-t (γ+(Sqrt[(γ-ω0) (γ+ω0)])))) (q0 ((Sqrt[(γ-ω0) (γ+ω0)])+(E^(2 t (Sqrt[(γ^2)-((ω0)^2)]))) (Sqrt[(γ-ω0) (γ+ω0)])-2 (E^(t (γ+(Sqrt[(γ^2)-((ω0)^2)])+I ωd))) (Sqrt[(γ-ω0) (γ+ω0)])+(-1+(E^(2 t (Sqrt[(γ^2)-((ω0)^2)])))) (γ+I ωd))+((-1+(E^(2 t (Sqrt[(γ^2)-((ω0)^2)])))) v0-x0 γ+x0 ((Sqrt[(γ^2)-((ω0)^2)])+(E^(2 t (Sqrt[(γ^2)-((ω0)^2)]))) (γ+(Sqrt[(γ^2)-((ω0)^2)])))) (-((ω0)^2)+ωd (-2 I γ+ωd))))/((Sqrt[(γ-ω0) (γ+ω0)]) (-((ω0)^2)+ωd (-2 I γ+ωd)))] ,{t,0,50}]
,{{x0,1},-10,10}
,{{v0,1},-10,10}
,{{ω0,1},-10,10}
,{{γ,0.1},-10,10}
,{{ωd,1.1},-10,10}
,{{q0,0.5},-10,10}
]