Equations of the First Order. 375 



Note on the application to ordinary Equations of the principle of 

 Duality established for partial Equations. 



19. For an exposition of this principle see Boole's Diff. Eq. 

 p. 366. Applied to ordinary equations of the first order, the 

 principle gives this proposition. 



If we have two ordinary differential equations, 



<£fc y> p)=®> <f>(p> p%—y> ®) =°> 



and either of them is solved by 



y — 'fXy 

 the other is solved by 



This principle has its most extensive application to the equa- 

 tion so often above referred to, 



y=xfp + <f>p. 



px—y =pfx + §x, 



y , fo } 



—fx x-fx 



p 



X 



Its dual is 

 or 



whose solution is 



f dx / C <kx f dx \ 



y = e J x-fx I c+ I r € J x-f*dx\ =\x (suppose). 



Consequently 



y = x\'~ 1 x — W'~ l x 



is the solution of the given equation. This form, however, often 

 involves the inversion of transcendental functions. If we take 

 y=\x, its dual is 



y=px—\p, or y—px + '\p = J 



in which we have merely to substitute for p its n values succes- 

 sively, and multiply together the n resulting expressions. 



Iffp=p (Clairaut's form), the mere process of transformation 

 gives an integral. For 



y —px = (f>p 

 transforms itself into 



y=<t>*> 



whence the solution 



y = X(f>'~ l x — <$><fi~ l x. 



As no arbitrary constant is thus introduced, it is obvious that in 

 this case the solution arrived at must be the singular solution. 



The transformation is applicable to all equations having alge- 

 braic coefficients. 



