Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONSBy demanding that the coefficients of each power of z vanish separately, we obtain thethree-term recurrence relation(n +2)a n+2 − 2na n+1 +(n − 2)a n =0 forn ≥ 0,which determines a n for n ≥ 2intermsofa 0 and a 1 . Three-term (or more) recurrencerelations are a nuisance and, in general, can be difficult to solve. This particular recurrencerelation, however, has two straightforward solutions. One solution is a n = a 0 for all n, inwhich case (choosing a 0 = 1) we findy 1 (z) =1+z + z 2 + z 3 + ···= 11 − z .The other solution to the recurrence relation is a 1 = −2a 0 , a 2 = a 0 and a n =0forn>2,so that (again choosing a 0 =1)weobtainapolynomial solution to the ODE:y 2 (z) =1− 2z + z 2 =(1− z) 2 .The linear independence of y 1 and y 2 is obvious but can be checked by computing theWronskianW = y 1 y 2 ′ − y 1y ′ 2 = 11 − z [−2(1 − z)] − 1(1 − z) (1 − 2 z)2 = −3.Since W ≠ 0, the two solutions y 1 and y 2 are indeed linearly independent. The generalsolution of the ODE is thereforey(z) = c 11 − z + c 2(1 − z) 2 .We observe that y 1 (and hence the general solution) is singular at z =1,whichisthesingular point of the ODE nearest to z = 0, but the polynomial solution, y 2 , is valid forall finite z. ◭The above example illustrates the possibility that, in some cases, we may findthat the recurrence relation leads to a n =0forn>N, for one or both of thetwo solutions; we then obtain a polynomial solution to the equation. Polynomialsolutions are discussed more fully in section 16.5, but one obvious property ofsuch solutions is that they converge for all finite z. By contrast, as mentionedabove, for solutions in the form of an infinite series the circle of convergenceextends only as far as the singular point nearest to that about which the solutionis being obtained.16.3 Series solutions about a regular singular pointFrom table 16.1 we see that several of the most important second-order linearODEs in physics and engineering have regular singular points in the finite complexplane. We must extend our discussion, therefore, to obtaining series solutions toODEs about such points. In what follows we assume that the regular singularpoint about which the solution is required is at z = 0, since, as we have seen, ifthis is not already the case then a substitution of the form Z = z − z 0 will makeit so.If z = 0 is a regular singular point of the equationy ′′ + p(z)y ′ + q(z)y =0538