100 INTEGRATION OF DIFFERENTIAL EQUATIONS. 



i 2 (n -p 



Now (n - p + A*) V 2 = a^" 1 (" -^ + 



By the application of this formula, y or 2& n # n may be de- 

 duced by a series of regular operations from C sin (yx + a). 



\ip is even, 2 (^> //,) + 1 gives the series 1, 5, 9, &c. 

 If it is odd, the series is. 3, 7, 11, &c. 



Particular cases may be solved by (22) with considerable facility. 

 By inspection we have 



( . 1 ) 



y = C -jsin (qx + a) H -- cos (qx + a) > 



for the solution of 



The solution of -j4 + 2 2 y = ? where p = 5, is easily .seen 



X* 



to be 



Lastly, the solution of 



s 



50 



^sin feaj + a) + ^ cos 



