98 INTEGRATION OF DIFFERENTIAL EQUATIONS. 



application it has an interest for some who care but little for 

 pure analysis, and because it will exemplify the general method. 



Let y = ^a n x n ........................... (2), 



... { % ( w _i)_6K + 2V 2 = ............... (3), 



n (n - 1) - 6 = n (n - 1) - 3 (3 - 1) = (n - 3) (n + 2) , 



... (w-3)(w+2K4V.M, = ............. (4). 



To get rid of the factor (n 3), assume 



(n-l)l> n ..................... (5), 



3 



= .................. (6). 



Hence b n is made to depend on b l or b Q as n is odd or even, 

 and we see at once that 



^b n x n = b cos qx + Z>j sin qx, 

 or changing the constants, 



2l> n x n = Csm(qx+a) ...................... (7). 



Also by (5), 



a = ^" 



for by (6), 



/. 2a B cc n = 2b n x n + 2 (w - 1) 



(9), 



the complete solution, which may be written thus, 



l.cos(^ + a)| ...... (10). 



