FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
277 
Referring to (2.), we have 
y(,(D)=D®; three roots equal to zero; 
fn(D) = q; 
w^A„.+^A^_„=0, law of formation. 
Hence the complementary series are, 
u—c 
jr 
t g g 
' rfi (2nY ‘ 
1_£ 9_ 
# (2n)^ (3w)^ 
g g g_ 
^3 
+ ...) 
+ C,|l0g ^ ( 1 - I + I - I (^3 ^ 
' G I (^3 ^ ^ (sIp • •) 
+ C3|(logx)"^l- ^ ^ 
g_ -r3n 
\3 
+ ..) 
(2ri)^ (2n)^ (3n)^ 
+ 6 log (i I (- + ^) J -(^3 (- + ^ + ^) ^ ^3 ^3 • .) 
' y ' \n^ ' {2nY ' n{2n) ) 7^ [2n)^ 
k ^ -h ^ 
, 3 
1 ^ ' 
\ g g g „3« , \ 
' (2n)‘-^ ' 
' {^nY * n{2n) 
~^n{^n) 
’"(2'ft)(3n)^ 
In' {2nY {3nY ' ' / 
and the particular solution is 
m=H — 9'j?”(D+w)“^H+^V”(Dd-w)"®(Dd-2w)“®H— ..., 
H being D~^(£*®G). 
15. The completeness of the preceding forms of solution depends, as above inti- 
mated, upon the circumstance that the function is of an order not lower than 
the order of the original equation. It may however be of a lower order, as would 
take place in the first of the examples above given, if instead of being 1 were zero 
and ^2 were also zero. 
Let the equation then be {Ex. 4.), 
Referring to (2.), we have, in the first instance, 
(D — 1 ) (£®m) + (c 2(D - 2) (D — 3) -h (D — 2) 4- flo) (f^u) = G, 
or 
DM+(c2(D-l)(D-2)H-Z»i(D-l)+ao)(£'’««)=s-®G; 
so that/o(D) is of the first order, and the root is zero. 
