The method used to solve simple differential equations is simple to that used to find the solutions to algebraic equations, with the difference that the coefficients ofare no longer constant but are functions of the independent variable. For each I we obtain a differential equation to be solved.

Example: Find a perturbation expansion up to the term infor the solution ofwith

Assume a solution

Substitution into the differential equation gives

Equating coefficients offor eachgives

(1)

(2)

(3)

The first of these, (1), gives the solution to the unperturbed equation,

Substitute this expression forinto (2) to obtain

This can be solved by the integrating factor method to obtain

Substitution of the expressions forandinto the third of these gives

This can again be solved by the integrating factor method to give

The complete solution is then