SYSTEMS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS. 



If we put rj = D/y , £= (D — \)z, we reduce (a) to the prime system 



T)x +v + £ =0 



(D-1)»+D,+(D-1){=0;; 



(D + 1>+D 2 ,,+(D + 1)£=0 



177 



08); 



for which 



K= D ,1,1 



D-l, D, D-l 



D + l, D 2 , D + l 

 -D(D-1)(D 2 -2D-1) 

 = -D(D-l)(D-a)(D-/3), 



y). 



where a = 1 + J2 , £ = 1 - V 2 

 The reciprocal matrix is 



-D 3 +2D 2 + D, o , D 3 -2D 2 -D 



D 2 -D-l , D 2 -l j -D 3 + D+l 



-1 , -D 2 +2D-1, D 2 - D +1 



(S), 



the columns of which are prime, except that corresponding to 77, which contains the 

 factor D — 1. 



The system (/3) is of the fourth order, and the forms in the complementary function 

 are 1, e\ e at , e 131 , the second of which does not occur in 77, owing to the presence of the 

 factor D — 1 in the corresponding column of (S). Hence the solution lias the form — 



r = a + b et+c <:M +d efi, 

 n =f +9 c at +h d 31 , 

 f = /•+/ c' + i» e at +nept} 



in which the four constants a, b, c, d may be taken as the four arbitrary constants of the 

 system (/3). 



By substituting the separate parts in (ft), we find at once 



/=«, k=-a, l=-b, g=2c, h = 2d, m=-(3+ >/2)c, n=-{3- J2)c, 



In the present case, since the particular integral isx , = 0,^ = 0,{=0, this, which is 

 merely a simplification of the ordinary method, is practically the quickest method of 

 solution. 



In order, however, to illustrate the general theory given above, we may give the 

 reduction to a diagonal system. 



Using the multiplier- system, 



D-l, - 1 I 

 D.-l I ' 



we can replace the first two equations of (/3) by two others, one not containing £, and 

 thus derive 



