PROFESSOE KELLAND ON GENERAL DIFFERENTIATION. 299 



_A,a-^_^B.r«'-i / xa'^-^ _ a' \ ^ _^ a' ^ X{x + 1) 

 ~a — 7 a — 7 + \ a — y {a — yf) ^ a^ + ^ a — y a'-^^ 



(y-af^ \ y, + J 



In precisely the same manner we may integrate the general equation with 

 constant coefficients. 



Let us apply the formula of Prop. 6 to the foUoAving example. 



Ex. 2. m^-3(j7 + 1)m^ + i+ 2(j' + l)(a' + 2)M,,,+o = 0. 



Calling (l+-\ / we get 



u, -SC^" +1) (1 + A^y«, +2 (e" +1) (e^ +2) (1 + A,)' ■•■ u, =0 



Now since /" (1 + AgYug =e~'' ' (1 + A^y e' " w^ 



we have 



or ug -3 (1 + AgYe^Ug + 2 (e" + 1) (1 + A^)'^^"^- ^ e" «^ =0. 



Put (i+A,y .(i+A,y for (i + A,)'^^""-^ 



where a^ in the former operates on the x in the latter. 



ug - 3 (1 + A^)' e'u^ + 2 (1 + e"") e (1 + A^)' . (1 + a^)'/ u, =0 

 or M^-3(l + A^)'e%(, + 2 (1 + A^)'/ .(1 + Arf)'e''«^=0 



or ug -3(1 + A^y e^M^ +2 (1 + A^y e^p m^ =0 



which can be resolved into the two 



{l-il + AtY e"}ii,=0 and{l-2(l+A^y/}?«^=0 

 or Uj.—{x + l)u,^i=0 and M.r— 2(a7 + l)%, + i = 



where ».=-^=t or "<•=; 



,X + 1 ' 2^';>+l 



A + B 2 ~ '' 

 and therefore generally, m,, = — , which is the complete solution of the equa- 



jx + l 



tion. 



It is evident that the process employed in Example 1 , applies equally in this 

 Example, when a function of x appears on the right-hand side of the equation. 

 Hence 



Ex.3. u,,-3(x + l)u^^i + 2{x+l)(x + 2)u^^2='^ gives 



