DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 347 



The system of equations^ (3) admits n linearly independent solutions, 

 such that, corresponding to each elementary divisor (r — r K ) eK of the 

 characteristic determinant, there are e K solutions : 



, „ * x r l -^ K 1 j ,, ~.i—\ / i = 1. 2. . . . n \ 



do) ,,-" = + 2 ^y, a^, (log .)' ^ = j; 2 ; J • 



( — A. 



.//V* ^s a multiple root of the characteristic equation which furnishes s 

 elementary divisors with the exponents e K , e.-^, . . . e. g _ 1 , then the 

 constants : 



A i, K ,e K > A i,K+l,e K+1 > • • ' J i,K+s-l,e K+s _^ (t = 1, 2, ... It) 



are s linearly independent solutions of the equations (4) when r = r K . 



§ 2. 



Solution op the Canonical System in the General Case 

 by Successive Approximations. 



We shall now return to the system of equations (1) ; and here again 

 we shall make use of the liuear transformation (10) to reduce the system 

 to the canonical form : 



d 1 r i ~ m " ?=e ' i ■ 



(16) ~dx Zki = x ***-* + i Zk ' 1 + 2 2 h Xt*u * 



i—l j=l 



(k = 1, 2, . . . m) (1=1,2,... e b ). 



The coefficients b' k j are linear functions with constant coefficients 

 of the coefficients a itJ in (1). 



We shall now make use of the method of successive approximations to 

 develop solutions of (16) about the point x = 0. It will be convenient 

 to write the equations (16) in the form : 



. .. i=m j=e,- 



(17) Tx ZkJ ~x ZkJ - x ~x Zi -' = 2, Z hftj- 



i=\ j—1 



The first approximation will be indicated by a third subscript 0, and is 

 obtained as a solution of the system of equations resulting from (17) by 

 making the right side zero: 



* This reduction is used by Sauvage in the case of a system of equations with 

 analytic coefficients. L. c, pp. 89, 90. 



