SYSTEMS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS. 175 



When the .system has a prime reciprocal matrix, and therefore admits of being 

 transformed into a simple diagonal system, the ordinary method of determining the 

 arbitrary constants by substitution is convenient. This method may be employed in 

 any case, and the work can be shortened by ascertaining from (30) what terms actually 

 do occur in the complementary functions of the respective variables. 



Other modifications suggest themselves in the light of the foregoing theory ; but it 

 seems unnecessary to pursue the matter here. Nor shall I enter into the interesting 

 question as to how far the general principles above laid down could be extended to 

 systems of ordinary linear equations whose coefficients are not constant. 



But it may be useful to append some simple examples to illustrate some of the points 

 of the general theory. 



Example 1 : — 



(D*+l) ^-f(D 2 + D + l) y = t, 

 Bx+ (D+l)y=e*. 



The characteristic determinant of the system reduces to 1 : we should therefore expect 

 that the solution contains no arbitrary constants at all. In effect, if we use the 

 multiplier-system, 



|1,-D I 



I D,-D--l|' 



whose determinant is constant, we deduce the equivalent system 



-y=i-2e', 



whence x = 1 + 1 - 3e\ y = 2e ( - 1. 



The fact that a system of differential equations may have a general solution which 

 involves no arbitrary constant whatever, is suggested naturally enough both by the 

 foregoing theory and also by Cauchy's theory of the order of any system. It lias, 

 however, been so seldom emphasised that it seemed worth while to give the present 

 simple instance of the phenomenon in question. 



Example 2 



Here 



(_3D 2 -4D + 1),+(D 3 + 3D 2 + 5D- l)y = 0f ' ' W " 



K 



D J -L> + 2, 2D-2 

 -3D 2 -4D + 1, D 3 +3D 2 +5P-1 

 S D(D+1) 3 (D 2 +1). 



The reciprocal matrix — viz., 



I'D 3 +3D 3 + 5D-1, 3D- + 4D-1 

 I -2D + 2, D 3 -D + 2 



