2. The main idea behind perturbation methods.

Consider the second order differential equation

where  is a small parameter,


We will try to solve it by inserting the perturbation series


and then find y0(t), y1(t), y2(t),…
We can for instance use

as a good approximation to the solution of the differential equation. Here y0(t) is the leading term, the solution of the unperturbed problem


The terms  are higher order terms which usually are “small”.

Remark: The unperturbed problem can in many cases be solved exactly.