
293Nonlinear Oscillations
© 2010 Taylor & Francis Group, LLC
xx
1
2
10
2
+=−−
(9.64)
xx xx
2
2
2201
2+=−−−ωββ
. (9.65)
The initial conditions are
which, in view of Equation 9.44, lead to
xAxi
ii
() () ,,,
0000 123== ==
.
From Equation 9.63, the zero-order solution is
x
0
= Acos ωt.
Substituting this for x
0
in Equation 9.64 yields
xxAtAt
At
1
2
1
22
1
2
1
1
2
21
+=−−
=− +−
ωωβω
ωβ
coscos
(cos ) A
co
ω
(9.66)
The cosωt term on the right-hand side (which already appears in the complementary solution)
generates the troublesome secular term. We now try to eliminate this term by setting the coef-
ficient of the cos(ωt) term equal to zero; hence,