Mathematical Methods for Physics and Engineering - Matematica.NET

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONSf(x) =0:a n (x) dn ydx n + a n−1(x) dn−1 ydx n−1 + ···+ a 1(x) dydx + a 0(x)y =0. (15.2)To determine the general solution of (15.2), we must find n linearly independentfunctions that satisfy it. Once we have found these solutions, the general solutionis given by a linear superposition of these n functions. In other words, if the nsolutions of (15.2) are y 1 (x), y 2 (x),...,y n (x), then the general solution is given bythe linear superpositiony c (x) =c 1 y 1 (x)+c 2 y 2 (x)+···+ c n y n (x), (15.3)where the c m are arbitrary constants that may be determined if n boundaryconditions are provided. The linear combination y c (x) is called the complementaryfunction of (15.1).The question naturally arises how we establish that any n individual solutions to(15.2) are indeed linearly independent. For n functions to be linearly independentover an interval, there must not exist any set of constants c 1 ,c 2 ,...,c n such thatc 1 y 1 (x)+c 2 y 2 (x)+···+ c n y n (x) = 0 (15.4)over the interval in question, except for the trivial case c 1 = c 2 = ···= c n =0.A statement equivalent to (15.4), which is perhaps more useful for the practicaldetermination of linear independence, can be found by repeatedly differentiating(15.4), n − 1 times in all, to obtain n simultaneous equations for c 1 ,c 2 ,...,c n :c 1 y 1 (x)+c 2 y 2 (x)+···+ c n y n (x) =0c 1 y ′ 1 (x)+c 2 y ′ 2 (x)+···+ c n y ′ n (x) =0.(15.5).c 1 y (n−1)1(x)+c 2 y (n−1)2+ ···+ c n y n (n−1) (x) =0,where the primes denote differentiation with respect to x. Referring to thediscussion of simultaneous linear equations given in chapter 8, if the determinantof the coefficients of c 1 ,c 2 ,...,c n is non-zero then the only solution to equations(15.5) is the trivial solution c 1 = c 2 = ···= c n = 0. In other words, the n functionsy 1 (x), y 2 (x),...,y n (x) are linearly independent over an interval if∣∣W (y 1 ,y 2 ,...,y n )=∣y 1 y 2 ... y n..′y 1′y 2... ....y (n−1)1... ... y n(n−1)≠ 0 (15.6)∣over that interval; W (y 1 ,y 2 ,...,y n ) is called the Wronskian of the set of functions.It should be noted, however, that the vanishing of the Wronskian does notguarantee that the functions are linearly dependent.491