DIFFERENTIAL EQUATIONS 207 



9.229 Many differential equations can be solved in a simpler form by the use 

 of the C n functions than by the use of Bessel's functions. 



(Greenhill, Phil. Mag. 38, p. 501, 1919) 



9.240 The differential equation: 



d?y 2(n+ i) dy 



M + '^r-^i 



with the change of variable: 



y = ZX~ n ~*, 



becomes Bessel's equation 9.200. 



9.241 Solutions of 9.240 are: 



2. y = *r- 



3. y = x--* H l n+ ,(x). 



4. y = *-*-* Hl +l (x). 

 9.242 The change of variable: 



X = 2\/Z, 



transforms equation 9.240 into the Bessel- Clifford differential equation 9.220. 

 This leads to a general solution of 9.240: 



When n is an integer the equations of 9.225 may be employed. 



sin (x + e) 



2 sin (x + c) cos (x + e) 



9.243 The solution of 



may be obtained from 9.242 by \yriting sinh and cosh for sin and cos 

 respectively. 



9.244 The differential equation 9.240 is also satisfied by the two independent 

 functions (when n is an integer): 



n sin a; 

 x 



S ( _ 

 - ,- - 4 _ o 



