If a system is perturbed by an amount
we can assume a solution of the form
and solve for the coefficients
to obtain the solution.
Example: Find a perturbation expansion for the solution to
Substitute
to obtain
Grouping together coefficients of
for each
gives
Then
Then
(1) can be solved exactly to give
and
The Taylor expansion of the first coincides with the perturbation expansion if
but the second root is
and is not present in the unperturbed equation
so cannot be given by the perturbed equation. We can find a perturbation expansion for the other root by putting
to obtain the equation
and assuming an expansion
Example: Find a perturbation expansion for the root near
of the equation
up to the term in
Assume
Grouping powers of
gives
Then