Finding Perturbations Expansions to Algebraic Equations
If a system is perturbed by an amountwe can assume a solution of the formand solve for the coefficientsto obtain the solution.
Example: Find a perturbation expansion for the solution to
Grouping together coefficients offor eachgives
(1) can be solved exactly to giveand
The Taylor expansion of the first coincides with the perturbation expansion ifbut the second root isand is not present in the unperturbed equationso cannot be given by the perturbed equation. We can find a perturbation expansion for the other root by puttingto obtain the equationand assuming an expansion
Example: Find a perturbation expansion for the root nearof the equationup to the term in
Grouping powers ofgives