April, 1916] Derived Solutions of Differential Equations 235 



Case VI. 



The non homogeneous linear differential equation. 



The solution is usually two or more series gotten from. 



y = x"^ (Ao+Ai, x^i+As x"2+A3X"34- etc.), (S), (where 

 A], A2, A3, etc., are functions of m, and each of all the preceding 

 by giving m particular values gotten by substituting x'" for y 

 in the given equation. 



Calling (S), y = F (m, x), then, in case of two equal roots 



^^"^'^^ y = -^F(m,x). 



dm 



a solution, if, after differentiation, m is given the value of this 

 root. And in case of three equal roots, is also 



a solution, and so on. 



Suppose k is a particular value of m, and that x"", Ai, A2, 

 etc., are all expressed in powers of m — k: 



x™ = xk (l + (m-k) logeX+ (m-k)^ log^eX+ etc.) 



Ao = Ao(l+Zero (m-k)+Zero (m-k)2+ etc). 

 Ai = Ao(ai+bi (m-k)+Ci (m-k)2+ etc.) 

 A2 = Ao(a2+b2 (m-k) + C2 (m-k)-+etc.), 



and so on. 



Substitute these values for the A's in S and calling 

 l + aiX"i-f-a2X"2-|- etc., the A-series; the coefffcient of m — k, the 

 B-series; that of (m — k)^ the C-series, etc., (S) becomes: 



y = Aox'^ (1+ (m — k) loge x+etc.) times 



(A-series+ (m — k) (B-series) + (m — k)- (C-series) 

 + etc.) = AqX*^ (A-series) 

 +AoX^ ( (A-series) loge x + B-series) (m — k) 



+ i\o x'^ ( ~ log2eX+ (B-series) logeX 



+ C-series) (m — k)-4-etc. 

 In case m has the value k only once, 



y = Ao x^ A-series (1) 



is the solution. 



If m has the value of k twice, 



y = BoX'' ( (A-series) logeX + B-series) (2) 



is also a solution. This is 



y = ^F(,n,x) 



when m = k. 



