[532 
TIAL 
7. (1869), 
itions by 
l on the 
îd in the 
îe second 
quadratic 
les of A ; 
number ; 
ir. Then, 
dear that, 
+k , ought 
arbitrary 
[ be the 
is when 
t between 
fractions 
532] ON THE INTEGRATION OF CERTAIN DIFFERENTIAL EQUATIONS BY SERIES. 459 
the numerators and denominators of which are factorial functions of a such that, for 
F K . 
some coefficient -j preceding - (if K is the coefficient of #“+*), and for all the 
G 
succeeding coefficients ^, &c. there is in the numerators one and the same evan 
escent factor; this being so, it is allowable to write F— 0, G = 0, &c. giving for the 
differential equation the finite solution 
( B E \ 
x a + ~ ¿» a+1 ... + Â ^ +<? J '■ 
but if, notwithstanding the evanescent factor, we carry on the series, then in the 
coefficient of x a+k there occurs in the denominator the same evanescent factor, so 
P 0 
that the coefficient of this term presents itself in the form A 77.x, = an arbi- 
trary constant K (since the - is essentially indeterminate), and the solution is thus 
obtained in the form 
B 
E 
y — A c ° a+1+ A xa+e ) + ^ [& a+k + jr x a+k+1 4- &c.), 
viz. there is one particular solution which is finite. 
Take for example the equation 
d*y dy 2 
¿ +q -£-^y =0 
a). 
mentioned Cambridge Math. Journal, t. 11. p. 176 (1840). If the integral is assumed 
to be 
y = Ax a + Bx a+1 + Cx a+ - 4- &c., 
then we find 
(a + 1) (a — 2) A = 0, 
(a — l)(a 4- 2)B + qaA =0, 
a (a + 3) C 4- q (a + 1) B = 0, 
(a + 1) (a 4- 4) D 4- q (a 4- 2) C — 0, 
&c. 
Hence a = — 1, or else a = 2 ; 
— qz. 
B = 
c= 
D = 
&c. 
A, 
(*-1)(x + 2) A ’ 
+ 1) 
(a — 1) a (a 4- 2) (a 4- 3) “ ’ 
— q 3 a (a 4-1) (a 4- 2) 
(a — 1) a (a + l)(a 4 2) (a 4" 3) (ct 4- 4) 
A. 
58—2