DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 



369 



We thus get the system of differential equations : 

 (84) 



dx = x '" + 



n 



x 



Vi 



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



The characteristic determinant of (84) is 



r — n + 1 + ft! fi 2 



— 1 r — n + 2 



(85) 





 





 







r— 1 



- 1 r 



The adjoints of the first line are : 



(86) r (r — 1) . . . (r - n + 2) ; r (r — 1) . . . (r — n + 3) ; . . . 



r (r — 1) ; r ; 1. 

 The characteristic equation is, then : 



(87) r (r — 1) . . . (r - n + 1) + fn r (r — 1) . . . (r - n + 2) + . . . 



+ /u„_ 2 r (r — 1) + /*„_! r + /*„ = 0.* 



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

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

 divisor of (85) corresponding to r K is (r — r K ) eK , where e K is the multi- 

 plicity of the root r K . 



We have seen that, corresponding to the elementary divisor (r — r K ) e " , 

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

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



(88) (r K -n+l+ ^) A^ K ^ + H A,^ + . . . + fi„ A nK ^ = 



-4-1^+ ('«-» + 0^,^ = 



(4 = 2, 3, . . . n). 

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

 ,qq\ A i,*,' K = pr,c(r K -l)...(r K -n + l + i) (i = 1, 2, . . . n - 1), 



A n , K,e K — Pi 



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

 vol. xxxvm. — 24 



