Poincaré-Lindstedt’s method is a method to avoid secular terms. Vi introducerar en “störd” tid
where
and put
Example: We examine Duffing’s equation
again. By doing the change of variables
the equation is transformed to
since
If we insert the the expressions for 
and u in this we get
and
Now we compare powers of 
:
 |
 |
 |
 |
By choosing
we see that we can avoid the secular term. This leads to
with the solution
Alltogether we have that a first order perturbation solution of Duffings equation is
where
Remark: Poincaré-Lindstedt’s method works for some (not all) equations on the form
Troubles arise when also the right hand side has the frequency 
in some step. We will then not succeed with the particular solution
but instead have to try with
which leads to secular terms. This trouble can be avoided in Poincaré-Lindstedt’s method by instead put
and then solve the corresponding equations as usual.