Due May 2
1. In class we developed the averaging method for describing nearly harmonic
motion for systems satisfying equations of the form ),(
2
0xxfxx DDD +−=
ω
where f is
assumed to be “small”. The approximate solutions were shown to be of the form
()
)(cos)()( 0tttAtx
φω
+= with
()
0
0)(sin
ω
φω
ttf
A+
−=
and
()
0
0)(cos
ω
φω
φ
A
ttf +
−=
where indicates averaging over one period. In this
problem I want you to use this method to get an approximate solution for a lightly
damped oscillator of the form xxx DDD
βω
2
2
0−−= .
2. Of course we can exactly solve the damped oscillator in 1. Show that the exact
solution agrees with the approximate one in the sense the frequency and damping
factor agree up to first order in
β
.
Problems 4.3, 4.4, 4.8