Consider a mass m that is suspended in a spring, where the restoring force F is supposed to be
in the distance y from the equilibrium point (with minus sign because the force is opposed to the direction of motion). Newton’s second law then gives
with the initial conditions (the mass is dropped from a positive displacement A)
We rescale this problem according to
(see chapter 1) and get the equation
where
is a dimensionless parameter. This is Duffing’s equation. We try to solve it with our perturbation method. Insert
in the rescaled differential equation:
Now compare powers of 
:
 |
 |
 |
 |
An approximate solution is thus
Note that:
(i) the leading term cos(t) seems correct.
(ii) if t<=T0 and is “small” then the correction term is “small”.
(iii) if we let t to be large (
) , the correction term can be large even though is small.
Remark: The problem in (iii) is due to the secular term
Our next move will be to introduce a method that avoids this problem.