# ws715 - Rob Bayer Math 55 Worksheet Homogeneous Linear...

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Rob Bayer Math 55 Worksheet July 15, 2009 Homogeneous Linear Recurrence Relations with Constant Coefficients 1. Find the general solution to each of the following recurrence relations: (a) a n +2 = 2 a n +1 + 3 a n (b) a n = 6 a n - 1 - 9 a n - 2 (c) a n = 4 a n - 1 (d) a n = 6 a n - 1 - 8 a n - 2 2. Find the solution to the recurrence relation a n +2 = 5 a n +1 - 6 a n ; a 0 = 3 , a 1 = 8 3. Find the solution to the recurrence relation a n = 6 a n - 1 - 9 a n - 2 ; a 0 = - 2 , a 1 = 6 4. A theorem from analysis says that r is a double-root of r 2 + br + c if and only if r is also a root of the derivative, 2 r + b . Use this fact to show that if r is a double root of r 2 + br + c , then nr n is a solution to the recurrence relation a n +2 + ba n +1 + ca n = 0 Non-homogeneous Recurrence Relations 1. Determine whether each of the following are homogeneous or non-homogeneous. (a) a n = 3 a n - 1 + 2 a n - 2 + n 2 (b) a n = a n - 1 - 4 a n - 2 + 1 (c) a n = a n - 1 + 7 a n - 2 + a n - 3 2. What is the “form” of your trial solution for each of the following. For convenience, the roots (and their
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