ω=5; 
δ=0.5
eqn1=y''[t]+2*δ*y'[t]+ω^2*y[t]==0;
eqn2=y[0]==1;
eqn3=y'[0]==0;

sol2=DSolve[{eqn1,eqn2,eqn3},y[t],t]

Plot[y[t]/.sol2,{t,0,10},PlotRange->{{0,10},{-1,1}}]