Characteristic Equation Method

Discrete Math · Axiom Academy

LESSON Characteristic Equation Method A systematic approach for solving linear homogeneous recurrence relations with constant coefficients using exponential solutions. Consider the Fibonacci recurrence: a_n = a_ n-1 + a_ n-2 . Rather than computing term by term, we guess that solutions have the form a_n = r^n for some constant r . When we substitute a_n = r^n into a recurrence relation, we get an algebraic equation in r called the characteristic equation . Step 2: Substitute to get the characteristic equation Step 4: Write the general solution based on root types Substitute r^n = 5r^ n-1 - 6r^ n-2 Divide by r^ n-2 : r^2 = 5r - 6 Factor: (r-2)(r-3) = 0 , so r = 2 or r = 3 When all roots are different, the general solution is a linear combination of exponential terms: Characteristic equation: r^2 - 5r + 6 = 0 gives r = 2, 3 Using a_1 = 0 : 2A_1 + 3A_2 = 0 When a root r appears m times (multiplicity m ), we need additional linearly independent solutions: Characteristic equation: r^2 - 4r + 4 = (r-2)^2 = 0 Double root: r = 2 (multiplicity 2) When roots are complex conjugates and , we can express solutions using trigonometric functions: Characteristic equation: r^2 = -1

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