138 A. Mukliopadhj^ay — Differential Equation to all Conies. [No. 2, 



If we assume -r-^ = z, the equation reduces to 

 ace 



oJae^ dx dx^ \dx) ~~ 



As this is homogeneous in z and its differential coefficients, if we put 



/ udx 

 z •=■ e"^ , 



the equation is transformed into 



9 -; ISu-rr+^u^ = 0. 



dx^ dx 



As this involves only the differential coefficients and the dependent 



, . . • . , ^^ d^'^ ^^ ^'^ 



variable, a legitimate transformation is to put —• z=z v, — — ^ = — = v — -, 



tt^ ax ax Cv'tw 



which give 



To separate the variables, assume 



whence — - = -w^ -; — |-2i*(X+|). 



ait du 



Substituting and simplifying, we have 



dX 2X - 6X^ 





du 1+3A ' 



where the variables are separated, 



du l + 3\ _. ( 1 . 6 ) 



Integrating, 



2 log hu = log X - 2 log (I - 3X), 

 where Tc is the constant of integration. 

 Therefore h^ u^ {1 ■- 2>Xy = \ 



which is the complete primitive required. It now only remains to 

 express u, X in terms of ^, y ; for this purpose, it will be convenient 

 to enumerate the successive transformations we have used, viz., 



dx 



From these we get 



dHi _ _ /^ 

 _ = . = (X+i)„.. 





