G14 Mr. Savidge on the Integration of a Class of 



The Generalized Equation. 

 Returning to the equation (1), 



t" + 2Ft' + (P 2 -i-F r -^i +.qx™- 2 -r\v 2 ™- 2 )t=0. (1) 



Putting *=fl£exp. {— §{F-\-rx m - 1 )dx}u 



and substituting, 



. . . . 2r 



and on putting in this equation X= — x m 



we obtain 



— ■ d 2 u / 1 \ du fa n 2 \ _ 



^ + Vx- 1 jrfx + Hx-x^ =0 ' 



where a = 77^— — ^ , n=^-. 



— zmr J. m 



This is the equation (2), and the solution of (1) may be 

 written 



* = ., Je *p.{ -JP.*-£}U*.(jJL -J; J>). 



The following are some of the principal forms comprised 

 in the equation. In the following h, c, f, g, h, k, I are 

 constants, in addition to m, p, 9, r. 



(1) t " + 2ct'-t\S^--(c 2 + q)+rV} =0. 



Solution : 



t=x* exp. (-«*- ^) U ^(£; - 2 ' n1 ' 2 )' 



( 2 ) *""+(^ + 2cx\t' + t { (6 ~*J~ 7 ' 2 + fe + 26c + c) + (c 2 -r 2 )x 2 1 =0. 



Solution : 

 t = xi-* exp. (- ^a^ij^QL - \ ; rx 2 }. 



(3) t" + (2foe+ 2c)*' - t(gx 2 + 2^ + *) = 0. 

 Solution : 

 *= (.»+/)iexp. { - 5+f (« +J F)i + J(„- + /) I Uj { a ; rtjc+ff } » 



