INTEGRATION OF DIFFERENTIAL EQUATIONS. 103 



It is obvious that analogous equations exist in all orders, and 

 that when p is of certain forms, 



may be integrated in finite terms. 



It will be sufficient, after what has been said for the cases of 

 m = 2 and = 3, to state the results of the general investigation ; 

 they may be very readily deduced by the same method as that 

 we have already used. 



The process succeeds when either of the factors p or p 1 is 

 divisible by w, and the general formula of which (22) and (26) 

 are cases, is 



...... (27). 



Particular cases may however be easily solved without re- 

 ference to this formula ; thus, if we had 

 d*y 4 _ 12 d*y 



dtf~zy-* dx *> 



we should proceed as follows : 



n(n-l){(n-2)(n-8)-4.8K- 2 V. = 0, 



n(n-l}(n- 6) (n + !)- tfa^ = > 



(n + l)a n = (n-2)l> n , 



/. n (n - 1) (n - 2) (n - 3) b n - gfb^ = 0, 



.'. ^b n x n = G/ x + C 2 e~ qx + C 3 sin (qx + a), 



O O / * \ j 



and a n =J tt - _.._ _ 



- - 4 ^n (n - 1) (n - 2) n aT 4 , 



+ <7 3 -jsin (^# + a) H -- cos (^a? -f a) [ . 

 ( g'a? ; 



