270 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



I See Art. 18.) For when 6=0, the equation becomes an ordinary linear equation 



1 

 of the first degree, of which the solution is y=cf-^. 



six 



In this case a=/3 and A= - B : 

 we may therefore write B= - A generally, and we obtain as the complete solution 



J- J- -i- _ — 



The above equation may be reduced differently, thus. The symbolical form 



/-D + 1 





may be written 



2 a {-D-i)y-h \/^-i. e~ - '- 



y-9:,«~''y=o, 



D " 2a 

 1 



or 



or 



h^ni -D -' 1 



y-- 



2a /-D + A 

 2a -D+* 



. e-^y=0, 



e '-y- 



1 /-D 



2a _D + i ;-D 





or 



y-- 



fcv^IIi (-D -^ 



-D 



2a 



-D+J 



e ^y- 



2a ._D. 



e - i/ = 0, 



-D + i 



which is of the form (1 — '-^ <Pi-w-„^x")y=^' 



of which the solutions are 



(1 + -^<^x) y = 0, and (1 + -|<^ J y = 0, or 



y^}-e'- -^^e-'y = ^, Wi^y + -^e- 



-D- 



/-D 



e-''y = 0, 



or 



rf-= .5' 



V— f-=-^- = 0, and y--^^ r 



= 0; 



which, on differentiation to the index h, give the same results as before. 

 Ex.2. y + a.vy + bx-^+2ax-'-^ = X. 



The solution is, as in Example 1, 



y = 



«-^^rw(^^«)-^^TT^(^^°) 



