POINTS OF THE INTEGRAL CALCULUS. [121] 



0„ = X, y\f n = F, so that F is implied in terms of X, when y is assigned in terms of x. Let 

 d Y : d X be F', &c. We have then 



$x . dy" = dX, ^x . dtf* - d Y, Y' = ^ , say x - x F. 



Consequently, /(<£„, ^„, a?) «= is reducible in the first instance to /(^F, K, ^F') = 0. If 

 from this we find Y= u(X, C), this, and x = yY' enable us to find X, and thence Y, in terms 

 of x and C. Substitute in <p n = X and >//■„ = F, eliminate y w , and we have a linear equation of 

 the order n - 1, involving the constant already obtained, and yielding n - 1 others in integra- 

 tion. 



a? 8 , 2d „ , 



For instance, let — y - xy +y ■ — (a?y - y ). 



2 a? 



Let -y" - «y' + y = F, *y" -ff-X, Y'= £*, 



Car 2 _ 4a C ... „ , 



X 2 = . F 2 = ; eliminate y and 



o(4a-^)' 4a-* 2 * 





whence the complete value of y can be found. 



By making 0„ = F, >//„_„,= X, whence T = 0a>.y ( " +1 >H- >//*>. y ( ''-'" +1, , we see that 



/ (*. *.-. *."£Z) = ° de P ends ° n /(^ F > r > = °- 



If this last give Y =sr(X, C), we have (p n = •ar(>//„_ m , C), which is one unit lower in ordei 

 than the given equation, and can be reduced in the manner already pointed out. 



We can make further discovery of reductions by using more differentiations. For example, 

 the last case but one gives 



F"d^=(^Vd*orF"= ^-Jtr)' 



Hence /{</,„, *. *, ^-1— . (£)] - 0, depends onf(X, F, x F, F') = 0. 



This last being integrated, we can express X by help of F', in terms of x and two constants ; 

 after which, substitution in the first equations, and elimination of y ( ">, gives a linear equation of 

 the order n — 1, from which the remaining constants are found. 



We may also, instead of using X and F, use functions of them similar to those already 

 given. For example, let 



2 



y"-xy'+y=XY'-Y, y" = \X* 



Then £a? 2 = F", — F" - XY' + Y = xy'-y, Y'"y'" = Xx ; &c. whence 

 Vol. IX. Paet II. 40 



