
10.18. Solve Equation 10.54 for Ω 1 using the approach suggested in the footnoted Equation 10.59.
Solution: Start with
θ
+ Ω
2
θ + εθ
3
= Γ cos τ.
Let Ω
2
= 1 + εβ and Γ = εγ, expand θ as θ(τ ) = θ
0
+ εθ
1
, and sub into the original equation
θ
0
+ εθ
1
+ (1 + εβ)(θ
0
+ εθ
1
) + ε(θ
0
+ εθ
1
)
3
= εγ cos τ.
Separating by terms produces
θ
0
+ θ
0
= 0
θ
1
+ θ
1
+ βθ
0
+ θ
3
0
= γ cos τ.
The solution for θ
0
is given by
θ
0
(τ) = b
0
cos(τ).
Using the initial conditions θ(0) = b
0
and θ
(0) = 0. This simplifies the equation for θ
1
to
θ
1
+ θ
1
= γ cos τ −β (b
0
cos(τ)) − (b
0
cos(τ))
3
= [γ −βb
0
−
3
4
b
3
0
] cos(τ) −
1
4
b
3
0
cos(3τ).
To avoid secular terms, choose b
0
such that
γ − βb
0
−
3
4
b
3
0
= 0.
The solution for ...