DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 



369 



We thus get the system of differential equations 



dyx ^ / ^i - w + 1 

 dx \ X 



(84) 



^Vi yi 



- + 7^2 \yi - 



The characteristic determinant of (84) is 



-[^ + P")y'- 



({ = 2, 3, . . . n). 



(85) 



r — n + 1 -{- fii //2 



— 1 r — 71 + 2 





 





 







r — 1 

 - 1 r 



The adjoints of the first line are : 



(86) r (r — 1) . . . (r — w + 2) ; r (r — 1) . . . (r — « + 3) ; . . . 



r (r — 1) ; r ; 1. 

 The characteristic equation is, then : 



(87) r (r - 1) . . . (r - M + 1) + /xi r (r - 1) . . . (r - w + 2) + . . . 



+ /"«-2 r (r — 1) + /x,,_i r + /x„ = 0.* 



There is always one first minor of (85) which is not zero, the adjoint 

 of fi„ ; and therefore if r^ is a multiple root of (87), the only elementary 

 divisor of (85) corresponding to ?\ is (r — r^Y", where e^ is the multi- 

 plicity of the root r^. 



We have seen that, corresponding to the elementary divisor (r — r^)**, 

 there are e^ linearly independent solutions of (84) of the form (80). 

 Here the constants Ai^K,e must satisfy the set of equations : 



(88) (r, - n + 1+ H-^) A,.,e, + /^2 A,.,6, + • • • + /^„ A,.,e, = ^ 



(^•=2, 3, . ..«). 

 This system of equations has essentially only one solution, namely : 



Ai, ,. . = p r, (r, - 1 ) . . . (r, - « + 1 + (i = \, 2, . . . n - 1), 



(89) 



-^«,»c,c^ — Pj 



* (87) is also called the indicial equation of (82) for the point x = 0. 

 VOL. xxxviii. — 24 



