1870.] Mr. W. H. L. Russell on Linear Differential Equations. 17 



in which the numerical coefficients are of no consequence, as the equation 

 does manifestly not admit of an exponential solution. If, then, the differ- 

 ential equation admits of a solution in the proposed form, it must be one of 

 the two forms, 



1 * 2 i 



y=R(aOe* +3 * + T l or y= R(»e~ 2 x ~\ 

 where U(x) is a rational and entire function of (a?) or a rational fraction. 

 Using the first form, we should of course determine it equal to oc-\- 1. 



As a second example, we form from the primitive, y — {oc— l)e the 

 equation 



(^-l)(^+l) 2 0+(2o; 2 +^-3)^-(^ + 5^ + 4^-f l)y=0. 



Here we must put y = R(#)e/^ x , higher powers of (x) in the exponential 

 leading to no result. Substituting, we find p= + 1. Let #== -—1, and 

 the differential equation becomes 



(2/+ . . . ) ff+(2* 5 + . . .)| +(* 3 + • • • )y=o, 



which gives y=H(z)e ss . If we put we find no exponential solution. 



Consequently the solution of the equation, if it can be obtained under 

 the form we are now considering, must be one of the two expressions, 



y=R(a?)e* *-i and y=R(#)e ,r 

 As a third example, I take the primitive, 



y=xe 



and from it the differential equation 



a? 2 



a? 



^ + (2a? + 2) ^-(^ + 4a* + 2a?-2)y=0. 



We must evidently here put y=~R,(x)efi xdx , which gives fi= + l. If 

 x— the equation becomes 



^ _ 2 2 5 ^ - ( 1 + 4z + 2z 2 - 2z 5 )y = 0. 

 dz dz 



If we put here y='R( t z)ef dz (i J - +l ' z \ employ the formulae given in the first 

 part of the paper, and equate the coefficients of the highest powers of (z) 

 to zero, we have v 2 — 2v=0, 2^—2^ = 0, whence v = 2, and ju = 0; and 



1 _ 4-1 



y must be one of the two forms I? (;c)e* + * 2 , or R(.r)e * 2 . 



VOL. XIX. 



c 



