460 
NOTE ON THE INTEGRATION OF CERTAIN 
[532 
Here taking a = — 1, we are at liberty to make C and all the following coefficients 
= 0: in fact, if w’e commence by assuming 
the equations 
y = Ax a + Bx a+1 , 
(a + 1) (a — 2) J. = 0, 
(a — 1) (a 4- 2) B + qaA = 0, 
qipL + 1) B = 0, 
are all satisfied if only a = — 1, B = 
solution y = A (i - |) 
indeterminate quantity 
— qaA 
and we have thus the finite 
(«_ l) (a + 2)’ 
solution y — A — 2) ’ but ^ we con fi nue series, retaining D to represent the 
-q 3 a(a + l)(a + 2) 
(a — 1) a {a + 1) (a + 2) (a + 3) (a + 4) 
A, we have the solution 
y = A ^ + D (x 2 - \qa? + &c.), 
the second member of which is in fact the series derived from the root a — 2. 
This series is expressible by means of an exponential, viz. we have 
* - *«**+ &c -=f {S+^ - (3 - fe)} - 
and the complete integral is thus 
y = A(l- iq ) + B(l+ to)**. 
but this result is not immediately connected with the investigation. It may however 
be noticed that, writing z = ye qx , the equation in z is 
d 2 z dz 2 _ 
dx 2 ^ dx x 2 Z ’ 
which only differs from the original equation in that it has — q in place of q: 
there is consequently the particular solution z = - + \q, giving for y the particular 
CO 
solution y = (- + e~ qx , and we have thence the complete solution as above. 
Consider, secondly, the differential equation 
d?y 
dx 2 
(II), 
(derived from the equation (I) by writing therein ye~ qx in place of y); this equation 
is satisfied by the series 
y = Ax a + Bx a+2 + Ox a+i + &c.