Repeated Roots

Combinatorics · Axiom Academy

Learn how to handle repeated roots in the characteristic equation and construct the modified general solution 1. The Problem with Repeated Roots Consider the recurrence relation with characteristic equation that has a repeated root r. If we tried to use the standard approach, we would get: But both terms are the same! We only have one independent solution, not two. We need a second linearly independent solution to form the general solution. The solution is to multiply by increasing powers of n. For a root r with multiplicity k, the k linearly independent solutions are: Each additional power of n creates a new linearly independent solution. Watch how these solutions grow differently: 3. Example: Double Root at r = 2 Let's see a concrete example. If the characteristic equation has a double root r = 2, then our general solution is: Notice how the second term has the extra factor of n, making it grow faster than the first term. For a characteristic equation with roots r_1, r_2, ..., where some roots may be repeated: Each root r_i with multiplicity m_i contributes m_i terms to the solution, multiplied by successive powers of n.

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