110 INTEGRATION OF DIFFERENTIAL EQUATIONS. 



/. a = mP -jj-jb n , and /. 2a n x m = m p -^ ^b n x n . 



The factor m p may obviously be neglected, and we shall 



j^ 

 therefore have, on replacing %b n x n by <f> (k) X, i. e. by -^-j , the 



Ji^ 

 following equation, 



d p X 



for the solution of (1), y = X being that of + ty = 0. 



Vfl "" 1 



If 7?i = 2, X= (7 sin {VW + }, and - = J. Hence 



d p Cs 



is the solution of 



Jl+%=^. 



rfa; 2 x dx 



This result is given in Hymers' Diff. Equations, and is, I believe, 

 due to Mr Gaskin. 



If the proposed equation were 



^ 



we should immediately conclude, from analogy, that its solution 

 must be 



_ dT* X f 



y T.-P -i ? 



but it may be as well to establish this conclusion by an inde- 

 pendent investigation. 



Equation (2) will, in this case, be 



n (n 1) ... (n m + 2) (n m + 1 +pm) a n + ka^ = 0. 

 Assume 



' a = f(1r\ _ n ~ 

 -'V ) 



