
10.8. Obtain an approximation to the forced response, of period 2π, for the equation
x
+
1
4
x + 0.1x
3
= cos τ.
Solution: For the system, we define ε = 0.1 for the nonlinear term, and for a periodic solution, we require
the condition x
i
(τ + 2π) = x
i
(τ) for all perturbation terms. Using the expansion x(τ) = x
0
(τ) + εx
1
(τ), we
substitute this back into governing equation
x
0
+ εx
1
+
1
4
(x
0
+ εx
1
) + ε(x
0
+ εx
1
)
3
= cos(τ).
Grouping the terms in terms of ε, we obtain
x
0
+
1
4
x
0
= cos(τ)
x
1
+
1
4
x
1
= −x
3
0
.
The solution to x
0
is then
x
0
(τ) = a
0
sin(τ/2) + b
0
cos(τ/2) −
4
3
cos(τ).
Due to periodicity, a
0
= b
0
= 0. If we do not set a
0
and b
0
zeros, we end up with resonant solution