178 



SYSTKMS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS. 



(D 2 -2D + 1).»- , =0 

 (D s - D + l)r>+0,,+f=0 



! 



By means of the multiplier-system, 



D + l, - 1 

 D, - 1 



for the last two equations of (e), we derive 





Finally, using 



(D 8 -2D+1)oj- n = 

 (D s -B)x -D 2 , ; = 



(D 3 - P 2 - 1) x - D'-'t? - f = 



I) 2 , - 1 

 D 2 + l, - 1 



on the first two equations of (£), we deduce the diagonal system 



( r)3_D 2 -i),-DW=o, 



(D , -? 1 D 3 +2D ! -D+l)x- ip ) 

 (D'-3D ;i +D 2 + D).r =0, 



which, by using the last equation in the second, may be reduced to 



1 



(D 3 -D 2 -l)ar-D 2 ^-^=0 

 (P*-2D + l)x- l} =0 

 (D 4 - 3 D 3 + D 2 + D) x = ) 



(e). 



(D- 



(v). 



The system (17) may now be solved very readily, and leads to the same result as 

 before. 



After solving (/3) by either method, we pass back to (a) by solving the two equations 



Dy=,,.CD-l)a=f; 



and thus get the complete solution of (a) involving six arbitrary constants. 



