532] 
DIFFERENTIAL EQUATIONS BY SERIES. 
461 
Then a = — 1 or a = + 2, but here the series belonging to a = — 1 contains only odd 
powers, the other contains only even powers of x; hence the two series do not 
coalesce as in the former case, and the first series is obtained without the indeter 
minate symbol in any of the coefficients. We have in fact 
(a + l)(a — 2) A = 0, 
(a + 3) (a. )B — lq"A = 0, 
(a + 5) (a + 2) G — \q 2 B = 0, 
and there is not in the series in question (or in the other series) any evanescent 
factor, either in the numerators or in the denominators. 
But consider, thirdly, the differential equation 
(derived from (I) by writing therein ye~ 6x in place of y). This is satisfied by the 
series 
y = Ax a + Bx a+1 4- Gx a+2 + Dx a+3 + &c., 
where a = — 1 or a = + 2 as before; in the series belonging to a. = — 1, the coefficient 
D should become indeterminate. The relation between the coefficients is here a 
relation between three consecutive coefficients, viz. we have 
(a + 1) (a — 2) = 0, 
(a + 2) (a — 1) .5 + (g — 20) ( a )A = 0, 
(a + 3) ( a )C + (q-20)(a + l)B + (0 2 -q0)A=O, 
(a + 4) (a + 1) D + (q — 20) (a + 2 )G + (0 2 - qd) B = 0, 
(a + 5) (a + 2) E + (jq — 20) (a + 3) -D + (0 2 — q0) G — 0, 
&c. 
It is to be shown that in the series for a = - 1, the expression (q - 20) (a + 2) G + (0 2 - q0) B 
contains the evanescent factor (a + 1), and consequently that D is indeterminate; we 
have in fact 
and thence 
(q- 20)(a + 2) G + (0 2 - q0) B 
u(ol + 3) 
1 
(g — 20) 3 a (a + !)(« + 2) 
(a + 2)(a+ 1) 
+ !) («+ 2) 20 ^ ^ ^ (a + 2 )l A - 
(i0- - q0) (q - 20) a 
(a + 2)(a-l)