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.