Ordinary Linear Differential Equations of 2nd Order, 611 



the series (3) to (6), that the asymptotic value of all finite 

 solutions is either u = x a x constant, or u = e x w~ a ~ 1 x constant.) 



^ i+ ^ + y "°y" in + --]- -•••■■• <»> 



L-tt-ifl ^~<« + l) a , 0r-{a + lV)(n*-{a + 2y) _ 1 



A comparison between these two series and the equa- 

 tions (3) to (6) and (20) shows that they satisfy the 

 following relations for all the finite solutions (including 

 those in terms of Bessel functions) : — 



cos(w — a)7r , 1 /OON 



Ti(ii + a) Y\(n — a — 1) 



1^= --^ -L— U g-ffiy T\ M * ' • V 24 ) 



II( — n-\-a) 1I( — n— a — 1) 



Second solution ivhere u 2 fails, — where In is zero or a 

 positive integer. 



If the equation u" + Yu\ + Qit — Q (P and Q functions of x) 

 has a known solution, say w 1? another solution is given by 



\j 



-J* 



A-^* (25) 



g ,?.dx 



Ui 



Thus when w=a we have a solution, on integrating by 

 parts : 



«, = - Mi e«(») + / "T 1 n( n~y 1} *-+• t. • (26) 



(the symbol of summation being subject to the convention 

 that as there are 2n terms under it there are none when 

 n=0). 



The corresponding solutions when n = a — 1, a — 2, &c, 

 may be found successively by application of the formulae 



* See Forsyth's * Differential Equations,' p. 110. 



J* 



(Phil. Trans, clx. (1870) p. 367). 



t The function E«(ff) or f— . civ has been fully tabulated by Glaisher 



