4.4.10  
Brannon and Boyce (baby 3rd ed.)

See corrected version; this contains several errors.

solution

lambda = -1/16;
omega = Sqrt[253/3]/16;

y[t_]=-(3/omega) E^(-t/16) Sin[omega t]

Simplify[16 y''[t]+16 y[t]/3+2 y'[t]]

y[0]
y'[0]

first pass through equilibrium

basicplot=Show[Plot[y[t],{t,0,50}],ParametricPlot[{Pi/omega,t},{t,0,3},PlotStyle->Dashed],Plot[0.01,{t,0,50},PlotStyle->Dashed]]

small amplitude

theta = N[Pi+ArcTan[-16 omega]]

atleast = (theta-16 omega Log[0.01/(3 Sqrt[3])])/Pi

j = 19
t1=(Pi-theta)/omega
t19=(19 Pi-theta)/omega

farplot=Plot[{y[t],-0.01,0.01},{t,90,110}]

Show[farplot,ParametricPlot[{t19,t},{t,-0.01,0.01},PlotStyle->Dashed]]