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 *y _{0}(t), y_{1}(t), y_{2}(t),…*

We can for instance use

as a good approximation to the solution of the differential equation. Here *y _{0}(t)* is the leading term, the solution of the

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

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