7. Poincaré-Lindstedt's method.

Poincaré-Lindstedt’s method is a method to avoid secular terms. Vi introducerar en “störd” tid




and put

Example: We examine Duffing’s equation


again. By doing the change of variables

the equation is transformed to




If we insert the the expressions for  and u in this we get




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



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.