Linear Differential Equations: 481 



Then by (C), 



D''M=(t!r-f A + r) .... (w + A„ + r)D^w„ 

 or 



D''w=(«r + A + r)-> (tT + ^„ + r)-^D''wi, 



and the remainder of the process as before. 



We will now illustrate this by two examples in each case. Let 



X = 7r(7r + a)w + A;(7r + 2V)(7r + ^>)a?'M. ... (3) 

 Assume 



Ms=(7r-f r)(7r-|-2r) .... (7r + 2V)wj, 

 then 

 af^u^x^inr + r) (7r + «V)wi = 7r(7r + ?') (7r+(i— l)r)a7''Wi 



by (A), It will be obvious that i is supposed to be an integer 

 number. Substituting these values, and operating with the in- 

 verse of the factors common to the second member, we have 



Xi = (tt + a)u^ + k{ir + ^x'^Uy^, 



Xj='7r-i(7r + r)-»....(7r+2V)-^X. 



This being only of the first order is immediately integrable. 



The next example is chosen because it cannot be reduced in 

 the same manner, and because it leads to a result of a very dif- 

 iferent form. 



Let X= 7r(7r— r)M + k{ir-\-a)(jr-\-a— (2i -f ly^x^^'u, (4) 

 Make 



w=(7r+a)(7r + a— 2r) (-Tr + a— (2i— 2)r)wi. 



Then by (A), 



^2r^--(^^^__2r) (TT + fl — 22V)a?2''Mi. 



By substitution and reduction as before, we have 



Xi = 7r(7r — r)wi + A;(7r + a— 2«V) (tt H- « — (22 + l)r)a?2''wi, 

 and 



Xi = (7r + «)-' .... (7r + a-(22-2)r)-'X. 

 If we put the last equation under the form 



X,= u, + ^(7rH-^~2fr)(7r-f^-(2i+l)r) ^^.^^^ 

 7r(7r — r) 

 it will reduce by (A) to 



^ \ IT / \ TT / 



Therefore, making 



(TT + a — 2i>\ 

 -^7— ;* =?' 



