DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 347 



77^6 system of equations (3) admits n linearly independent solutions, 

 such that, corresponding to each elementary divisor (r — r^Y'^ of the 

 characteristic determinant, there are e^ solutions : 



(10) y, -X 2(^-A)!^^.'^.'^'°S^^ \X = \,2,...e. 



I — A 



If Tk is a multiple root of the characteristic equation which furnishes s 

 elementary diinsors with the exponents e^ , e^ , ^ , . . . e^,^_^ , then the 

 constants : 



A,K,e^' A,K+l,e^^l^ • ' ' ^i,^+s~l,e^_^_,_i^ (i = 1, 2, . . . n) 



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



§ 2. 



Solution of the Canonical System in the General Case 

 BY Successive Appuoximations. 



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

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

 to the canonical form : 



(16) -j''^'^.^ — ~ ^-^.'-1 + — ^-t,/ + 2 2 ^'/i^i.j * 



ax X X r^ ^ '' ■' 



(^=1, 2, ...m) (Z= 1, 2, . . . e^). 



The coefficients V/f^^ are linear functions with constant coefficients 

 of the coefficients a, ^ in (1). 



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

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

 to write the equations (IG) in the form: 



t^n\ d \ ^^ V V l'-^' 



^^^ Tx''''-'~x'''''-^~'x^''''^ -^ 2.*.-./^0- 



t=l 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 tlie right side zero: 



* This reduction is useil by Sauvage in tlie case of a system of equations with 

 apalytic coerticieiits. L. c , pp. 89, DO. 



