DUNKEL. LINEAR DIFFERENTIAL EQUATIONS. 



343 



in the equations (3), and then determine the constants r and C^ so that 

 the equations are satisfied. We obtain in this way the following system 

 of n homogeneous linear equations for the C's : 



(4) At,,i Ci + ... + (/.,,, -r) a + -.- + /^,.« a, = (^•=:l,2,...n). 



The necessary and sufficient condition that (4) may be satisfied by a 

 set of C's not all zero is that the determinant: 



(5) 



A(r) 



/^i. 1 



\^n,\ 



/^l. « 



i"-«, n 



shall vanish. This determinant equated to zero gives an equation of the 

 wth degree in r, wliich is called the characteristic equation of (3) ; * the 

 determinant itself we may call the characteristic determinant of (3). 



If the characteristic equation has n distinct roots r^, r.^, . . . r,^, we can 

 determine n linearly independent solutions of (3) of the form : 



(6) 



V = C x-> 



i = 1, 2, 



1,2, 



. n 

 . n 



If however there is a multiple root, there will be, in general, solutions 

 involving powers of log x. To determine these solutions, we must 

 examine the minors of the characteristic determinant (o), and ascertain 

 if this multiple root is also a root of all the first minors, second minors, 

 etc. For the further stuily of this case, it will be useful to introduce the 

 conception of the elementary divisors of A(r). 



Suppose r' is a root of the determinant A (r) such that all the jt>th minors 



of A(r) are divisible by (r — ?•')'*, but no higher power of r — r divides 



them all. In the same way (r — j')"'"*'^ shall be the highest power of 

 r — r' dividing all the (jo + l)th minors. Then the expression : 



(r - r')'f 



ip — If If+i 



* Sauvage : /. c. p. HO. 



