210 



Mr. W. H. L. Russell on Linear 



[Feb. 10, 



III. "On Linear Differential Equations."— No. It. By W. IT. 

 L. Russell, F.R.S. Received January 20, 1870, being made 

 up of two Papers received December 30, 1869, and January 6, 

 1870. 



The principles laid down in my former paper will enable us to integrate 

 a proposed differential equation, when the solution can be expressed in the 

 P 



form pre w , where P, Q, w are rational and entire functions of (x). 

 Let 



(« 0 + %p + cc 2 x 2 +...+a m x m ) £| + 



(ft+A W+-..'+A^ B, )£^+ 

 (y 0 +yi*+y^+ • • • +y m * m ) + ■ • • 



+ (X 0 + X 1 + X 2 * 2 + . . . +X m ^)y = 0 



be the general linear differential equation of the ?iih order, where none of 

 the indices of (x) in the coefficients of the succeeding terms are greater than 

 those in the coefficients of the two first. Then if the equation admit of a 



solution of the form eJ <Pi(*) , where ^(a?), 0 2 (#) are rational and entire 

 functions of x, and ^(x) and o^ + a^ . . . + a m x m have no factors in com- 

 mon, and if the degree of the coefficients of the two first terms is the 

 same, 



2/=E(#) e-fp dx , 

 where p is a rcot of the equation 



pX-p^+r 2 /.. • • ±X W = 0; 



and if a m =0, 



and if a m = a m _i = 0, 



rj Tm , « m _ 3 0 m _i , m-fi m "w-^ro "m-M , £,„ 

 —fdx< — 1 4 +/ = J*H 



« ft. % _ 2 « 2 m _ 2 j -»-. 



and so on, where the value of ?/ is to be substituted in the proposed equa- 

 tion, which then becomes a linear equation to determine the rational and 

 entire function E(a?). 



When, however, a TO =/3 m =0, or, in other words, when the degree of the 

 coefficients of the succeeding terms of the proposed equation exceeds the 

 degree of the coefficients of the two first, some modification is required ; 



