200 Mr Glaisher, On the developments of [Nov. 24, 



Solution of differential equations by series, § 22. 



§ 22. In the solution of differential equations by series it is 

 well known that, supposing the equation to be linear and of the 

 second order, if we obtain as particular integrals two series R and 

 S, then the general integral of the equation is 



u = ^E + c^S; 



but that if in the formation of the series S the coefficient of a 

 term becomes infinite, the general integral is of the form 



u = aE log bx + T, 



T being a new series, and a and b being the arbitrary constants. 

 The seven differential equations afford examples of this principle. 

 Taking, for example, the fourth equation of § 17, viz. 



d 2 u 

 4A, (1 — ft) -^ — m = 0, 



and, following the usual process, let 



u = tA r K n+ \ 



the summation extending to all positive integral values of r ; 

 then, substituting in the differential equation, we have 



4 (m + r) (m + r - 1) A r - (2m + 2r - 3f A r _ x = 0, 



whence, putting r = 0, we find 



m = or 1. 



The equations giving A lt A 2 , A z ... are 



2m (2m + 2) A 1 - (2m - l) 2 A = 0, 



(2m + 2) (2m + 4) A 2 - (2m + l) 2 A x = 0, 



(2m + 4) (2m + 6) A 3 - (2m + 3) 2 A 2 = 0, 



&c. ' &c. 



Taking the root m = 1, 



r 



A — A 



1 2.4 °' 



2 4. 6 l ' 



A=^A 

 6 8 



&c. 



