delta=1;

w=10;
eqn1=y''[t]+2*delta*y'[t]+w^2y[t]==0;
eqn2=y[0]==1;
eqn3=y'[0]==0;

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

Plot[y[t]/.sol,{t,0,3},PlotRange->{{0,2},{-1,1}},
 AxesLabel->{"t[s]","x[m]"},LabelStyle->Directive[Large]]

delta=10;
eqn1=y''[t]+2*delta*y'[t]+w^2y[t]==0;
sol2=DSolve[{eqn1,eqn2,eqn3},y,t];
delta=20;
eqn1=y''[t]+2*delta*y'[t]+w^2y[t]==0;
sol3=DSolve[{eqn1,eqn2,eqn3},y,t];
Plot[{y[t]/.sol,y[t]/.sol2,y[t]/.sol3},{t,0,3},PlotRange->{{0,2},{-1,1}},PlotStyle->{Red,Blue,Green},AxesLabel->{t,x}]