TRANSACTIONS OF SECTION A. 469 



14. On the Solution of a Differential Equation allied to Biccatis. 

 By J. W. L. Glaisher, M.A., F.B.8. 



The equation referred to was the well-known differential equation 



(Pu i (i+l) 

 dx*~ ~!?~ u - t 1 ' 



which is transfonnahle into Eiccati's equation 



^' W 1 -2,7-2 



me 



by the substitutions 



1 



u = x~'tc, x = -s», 

 q 



where q = ^y 



The author stated that he had found that the differential equation (1) was 

 satisfied by the coefficient of h i+1 in the expansion in ascending powers of x of the 

 expression 



ga\f(x*+xh) 



so that the complete integral of (1) was 



u = A . coefficient of h i+1 in e a ^^+ xh) 

 + B . coefficient of #+' in e - a ^ +xh) . 



To prove this, consider the partial differential equation 

 A particular integral of this is 



■y _ e a*J(.x" + xh) 



for from this value of v we obtain at once by differentiation 



tPv „ (x + hhY w 



ax* x* + xh (x' + xhyi 



dx 2 ~ av ~x' i dh' i l ' 



(Pv 



■a'v — s r — av- 



then 



d 2 v 



dh* ~ " " .r 2 + xh ~ " " (x* + xh) h 



i satisfied. 



f v be expanded in ascending powei 



e oVc«»+*B = P + PjA . . . + P,A' + P,- +1 * i+1 + Sec., 



whence we see that (2) is satisfied. 



Let the above value of v be expanded in ascending powers of h, so that 



■ ■*inEr- o , p«.,)» w + &c. 





jo — n 2 y = 

 oar 



x*dv =.•••+ 4^ p ^ ,+1+&c -' 



and therefore, P, +1 satisfies the differential equation 



<Pu , t'(i + l) 

 ^ - «-« = -?-«■ 



In this solution it has been assumed that i = a positive integer ; but if i be nega- 

 tive = - »' — 1, we have i'(« + 1) = (*' + l)t', so that we may replace i by »', which is 

 positive ; and thus the solution applies when i = + an integer, corresponding to tho 

 integrable cases of Eiccati's equation, viz., when q = + the reciprocal of an uneven 

 integer. By considering the coefficient of h i+l in the direct expansion of 



goVfcrM-xA) . 



a 2 

 = 1 + o(.v 2 + xh)* + y~2 (** + xh ) + &c -> 



