Ordinary Linear Differential Equations of 2nd Order. 61 3 



Relations between u 7 , u s and the other solutions. 



A comparison of u 8 with u 5 for all finite solutions shows 

 that 



% 



cos znir 



-• n(-n-a-l) *' 

 or, what is the same thing, 



7rcosec(?i + a)7r Q /9Q v 



m 8 = =7-^ — ^— cos 2n7T . m 5 . . . (jy; 



11(71 + a) 

 A] o, by comparing the coefficients o£ #" in w l5 w 7j and w 8 , 



M 7 =?i 8 + M 1 {7T COt (?i + a)7Tj . . . • (30) 



From these two we obtain 



w 7 = — 



7rcot (n — aW 7rsin(n + a)7T /onN 



^ttT TV w 6 ttT — i — \ cos 2n7r ' w 5- Col) 



11 (?z — a — 1) II(« + a) 



When (a — n) is a positive integer the last reduces to 



u 6 cos (n-a)7r= - n ^l^ ) - • • • ( 32 ) 



It should be noticed that in some cases the solutions w 5 and 

 u 6 (21), (22) give at once two simple finite solutions equi- 

 valent to the above results. 



The importance of these relations lies particularly in the 

 fact that unless suitable solutions are selected, one degen- 

 erates virtually into a constant multiple of the other, the 

 distinct solution appearing only as a relatively small error. 



Solutions in terms of Bessel functions. [See (20).] 



The following are the solutions in terms of Bessel 

 functions when a=— |, 0, J, 1, and — 1. The second 

 solution is the corresponding expression having K B (#) for 



Utt(i ^) =^^ ) ^C* 1 -i(W2)-(2n-l+#)].(*/2)], 



L(l ; «) = ^^t* 1 ^' 1/2 [(*+n- 1) W«/2) + (ti + 1-«)I.-*W»)1 



QSn-lTTfo, n 



U„(.-l ; *) = fjjgs gJ «*«W[]^(«/2) + I» +i W2)]. 



