Download Theorem [On Solving Certain Recurrence Relations]

Theorem [On Solving Certain Recurrence Relations]
Consider the second order homogenous linear recurrence relation.
f(n+1) +a f(n) + b f(n-1) = 0
where a and b are constants.
(*)
Consider also the quadratic equation x^2 + ax + b = 0. [This equation is called the
characteristic equation of the relation (*).]
CASE I: The characteristic equation has two distinct real roots, R and S.
Then:
(a) The function R^n is a solution to (*).
(b) The function S^n is a solution to (*)
(c) EVERY solution to (*) is of the form:
P (R^n) + Q (S^n) for some constants P and Q
CASE II: The characteristic equation has one repeated real root R.
Then:
(a) The function R^n is a solution (*)
(b) The function nR^n is a solution to (*)
(c) EVERY solution to (*) is of the from
P (R^n) + Q (nR^n) for some constants P and Q.
CASE III: The characteristic equation has two distinct complex roots
u+iv and u-iv, where u and v are real numbers.
Then: letting theta = arctan (v/u) and
and letting T = square_root (u^2+v^2),
(a) T^n sin (n theta) is a solution to (*)
(b) T^n cos (n theta) is a solution to (*)
(c) EVERY solution to (*) is of the form
P [T^n sin (n theta)] +Q [T^n cos (n theta)] for some constants P and Q.
Furthermore, there are no other possibilities. That is, every quadratic equation
x^2 + ax + b = 0 either has two distinct real roots, one repeated real root or two complex
roots of the form u+iv and u-iv.
COROLLARY: The solution to the predator-prey equations are…… (we'll do that next).