17(5 



PROFESSOR CHKYSTAL ON THE EQUIVALENCE OF 



has all its columns prime ; hence any equivalent diagonal system must be simple. If we 

 take x as the last variable, the matrix of the system of multipliers will be 



L, M, 



D 8 +3D 2 +5D-1, _2D + 2 ' 



where L and M must be so determined that 



L(21 ) - 2) + M(D 3 + 3D 8 + 5D - 1) = const. 



The quickest method in practice for determining L and M will be understood from 

 the following special application : — Let 



fc- = D 3 +3D 3 +5D-l 



Eliminating D 3 , we get 



(0). 

 (7). 



(S). 



1)^-2/ = -8D 2 -10D + 2 



Next, eliminating D 2 by means of (/3) and (8), 



(I)- + 4D)^f-2/-= -18D + 2 (e). 



Finally, eliminating D by means of (/3) and (e), 



(D*+4D + 9)u-2v= -1G; 



we may therefore take L = D 2 + 4D + 9 and M = -2, so that the required multiplier- 

 matrix is 



D- + 4D + 9, -2 



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



This reduces the system (a) to 



-1G?/ + (1) 5 + 4D 4 + 8D 3 +4D 2 + 7D + 16>; = 6«- 

 D(D + l) 3 (D 2 +l)r = -4c-'; 

 n simple system as predicted. 



From this we get by the familiar methods, 



x=A+(B + Ct+Et 2 )e- t + ¥ cost+Gcsmt+^fe-' , 

 and, substituting this value in the first equation, 



y = A+H(2B + 3C-3E)+(2C+6E)*+2E* 2 }e-' 



+ F ( ., )S / + Q sin t+ 1(- 1 - 6^ + Qt*+ tfy- 1 . 



Lte+Py+(D-1>=0, 

 (D-l)x+V*y+(D-l) 2 z=0 } 

 (D + l)x+ Jf'v + (I )" - 1 >- = 



Example 3: — 



