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Finding Solutions Using Perturbation 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

Substituteto obtain

Grouping together coefficients offor eachgives

Then

Then

(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

Assume

Grouping powers ofgives

Then