Mathematical Methods for Physics and Engineering - Matematica.NET

15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTSthat a particular solution of the form u n = Aα n should be tried. Substituting thisinto (15.26) givesAα n+1 = aAα n + kα n ,from which it follows that A = k/(α − a) and that there is a particular solutionhaving the form u n = kα n /(α − a), provided α ≠ a. For the special case α = a, thereader can readily verify that a particular solution of the form u n = Anα n is appropriate.This mirrors the corresponding situation for linear differential equationswhen the RHS of the differential equation is contained in the complementaryfunction of its LHS.In summary, the general solution to (15.26) isu n =with C 1 = u 0 − k/(α − a) andC 2 = u 0 .{C1 a n + kα n /(α − a) α ≠ a,C 2 a n + knα n−1 α = a,(15.27)Second-order recurrence relationsWe consider next recurrence relations that involve u n−1 in the prescription foru n+1 and treat the general case in which the intervening term, u n , is also present.A typical equation is thusu n+1 = au n + bu n−1 + k. (15.28)As previously, the general solution of this is u n = v n + w n ,wherev n satisfiesv n+1 = av n + bv n−1 (15.29)and w n is any particular solution of (15.28); the proof follows the same lines asthat given earlier.We have already seen for a first-order recurrence relation that the solution tothe homogeneous equation is given by terms forming a geometric series, and weconsider a corresponding series of powers in the present case. Setting v n = Aλ n in(15.29) for some λ, as yet undetermined, gives the requirement that λ should satisfyAλ n+1 = aAλ n + bAλ n−1 .Dividing through by Aλ n−1 (assumed non-zero) shows that λ could be either ofthe roots, λ 1 and λ 2 ,ofλ 2 − aλ − b =0, (15.30)which is known as the characteristic equation of the recurrence relation.That there are two possible series of terms of the form Aλ n is consistent with thefact that two initial values (boundary conditions) have to be provided before theseries can be calculated by repeated use of (15.28). These two values are sufficientto determine the appropriate coefficient A for each of the series. Since (15.29) is499