DERIVATIVES ASSOCIATED WITH LINEAR DIF1 I.HKM IAI, IMITATIONS. 419 



SECTION IV. 



ASSOCIATE EQUATIONS AND DEPENDENT VARIABLES. 

 LAQRANUE'S " Equation adjointe." 



52. It was proved by LAORANGE,* that in connexion with every linear differential 

 equation there exists another linear equation of the same order, and that a know- 

 ledge of the primitive of either is sufficient to lead to the primitive of the other. Let 

 y\,y*, , y be n special and linearly independent solutions of the equation 



then 



,1.1" 



v = v, = 



<-*> 



y - 1 



y i 



y i 



is an integrating factor. For, since y\,y$. >y-\ satisfy the equation separately, 

 the 7i l quantities R can be found in terms of them ; and, when these values of K 

 are substituted and the equation is then multiplied by v, it takes the form 



,/., 



y . y . 



(n - 1) (n - 2) 



y._3, y.- 2 , 



y 



y -: 



= 0. 



But an integrating factor of the equation satisfies the relation 



* ' Miscellanea Taurinensia,' vol. 3, 1762 ; ' OZnvres,' vol. 1, p. 471." Solution dc differcnts problt-nu s 

 de calcul integral." 



8 H 4 



