496 Rev. B. Bronwin on the Integration and 



d 2 u , du , 



-t—c> +KX-, (p— l)fcu = Q. 



dx l dx l ' 



This is similar to the proposed, p being diminished of unity. 



By p operations, therefore, the last term will be taken away. 



Hence if y— (D + k x)p u, 



A d 2 u k du 



then + Jc 2 -o 



dx 1 d x 



an integrable equation. 



To take a third example, let 



Make y= D 3 u + k 3 u = (D 3 + P) u. 



We have 



x j^ = x D 6 u + ^ x D 3 u = D 6 # ?/ - 6D 5 w + F D 3 ^ - 3* 3 D 2 w 



= (D 3 + F) (D 8 tf«-6D 2 M ) +3FD 2 «, 

 and 



k 3 x y=Is? x D 3 u + k? x u= P D 3 x a— 3FD 2 « + 1^« 



= (D 3 + F) F«M-3FD 2 a. 



With these values the given equation becomes 



(D 3 + P) {D 3 .r u+ (3 p-6) D 2 u+A. 3 x u} = ; 



or D 3 ^w+ (3^— 6) D 2 e< + F *?*=(); 



X {d^ + Pu ) +3 ^-^dx^ = °- 

 Therefore if y = (D 3 + /r*)£ u, the proposed reduces to 



which we know how to integrate. 



Next, let x - r ^ + ( ( 2p-\-2mx)(-^ + «V] = 0. 

 ax 4 * M \a.v / 



Here we make y= (D 2 + 2 m D + 2 m 2 ) u. 



d z v 

 Then # -^ =.r D 4 m + 2 w.r D 3 « + 2 ra 2 .r D 2 « 



= T> 4 xu— 4 D 3 « + 2 »* D 3 a? m— 6 m D 2 m + 2m 2 D 2 ^— 4m 2 Du 



= (D 2 + 2mD + 2m 2 ) (D 2 i«-iD«) + 2mD 2 «+l»! 2 DM 



2 m x -r- =2»2«;D 3 « + 4 w 2 .r D 2 « 4- 4 w 3 a? D w 

 dx 



= 2 mD 3 x u — 6 mD^n + ^m 2 T) 2 xu — 8m 2 Dw + 4m 3 D#tt — 4m 3 « 

 = (D 2 +2OTD + 2m 2 )(2»jD^w-6mw) + 4m 2 DM +8 m 3 u 



