INTEGRATION OF DIFFERENTIAL EQUATIONS. 109 



If we substitute these values in (2), every factor of (4) and 

 every factor, except the last, of (3), will disappear, and the re- 

 sulting equation will be 



n(n-l)...(n-m + 2)(n-m + l)l n + kb n _ m = Q ...... (5). 



This is what (2) would be, were p = 0. Hence y = 2 . b n x n 



d m y 

 fulfils the equation ~^ + Jcy = 0, to which (1) would, in that 



case, be reduced. Let y X be the ordinary form of the solu- 

 tion of the last-written equation. Then we must obviously 

 have 



where (f> (k) may be any function of Jc. A very little attention 

 to the mode of integrating linear equations with constant co- 



efficients will show, that in X, x always occurs in conjunction 



j 

 with 7c m . 



If we put 



we must consequently have 



A n = 



where N is a function of n, except for values of n < m, when it 

 is an arbitrary constant ; therefore 



b n = N<t> (K) 7c. 



Recurring to (3), inverting the factors and multiplying and 

 dividing by m p , we shall easily deduce the following equation, 



m-l 



m 



The form of these p decreasing factors naturally suggests the 



idea of making f(Jc) k~ p ; and if we then put <f> (Jc) = Jc 

 we get 



m+l 



l\ fn 

 j\ 



mm 



