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LXVIII. On the Integration of a Class of Ordinary Linear 

 Differential Equations of the Second Order. By Harold 

 G. Savidge *. 



THE class of equations to be considered may be expressed 

 by the generalized equation : 



dH 

 d 



h^rA^f,-^^' 2 -^- 2 }^ CD 



where m, p, q, and r are constants and P a function of x. 



For the present we shall confine our attention to the 

 following equation : 



dashes denoting differentiation with respect to x, and a and 

 n being constants. 



We shall define JJ n (a ; x) as a solution of this equation, 

 and it will hereafter be shown how the equation (1) can be 

 solved in terms of this function. 



Formal solutions of equation (2) in convergent series of 

 ascending powers of the argument can be obtained for all 

 values of the constants a and n ; in particular finite solu- 

 tions in terms of known functions can be obtained in the 

 majority of those cases which will most frequently arise in 

 practical problems ; and the main object of this paper is 

 to show at once the nature of these solutions and the substi- 

 tutions in the variables necessary to lead to them. The 

 series solutions can be written as follows : — 



ll ' U f"lj_ n ~ a x i (n-aXn—a + l) .r 2 1 ,„x 



H(2n)L 2^ + 1* i (2^+l)(2n + 2) *|2 " t "** ,, J " " * w 



„ - x ~ n [~1 a. n + a £ -j- fr4-g)(tt + g-l) x 2 1 ,. 



2 ~n(-2n)L + 2n- l'|l + (2n-.l)(2»-2) '\T + ""] ' ' W 



and (by the substitution u = e x v) 



(= \_ ® n e x f.n + a + l £ (w + a + l)(w + « + 2) ^_ l 



M- u i) ~ n(2%) [1- 2n + 1 - • |]_ + (2?i + l)(2n + 2) • |2_ ' ""J 



... (5) 

 / n_ #~V r T n — a — 1 x (n — a — l){n — a — 2) x 2 

 UA ~ U ' 2) " n(-2«) L 2^T ' fl_ + (2n-l)(2n-2) • jl + ' ' * * J 



■ • • (6) 

 * Communicated by the Author. 



u,= 



