] 870.] Differential Equations. 21 1 



thus if 



(«+/k)§+(*'+/T*+y'*0g +(a"+/3"^y"r + 2Y + rV) g 



+ (a'" + 0"'* + y' V 2 + 2" V + C'"^ 4 + *" = 0, 



y = E(.v)e-/ ir tr " + <T' 3 "T 7 + r ' 

 where E(V) is to be determined as before. 



But now let <p L x and a 0 +a x #-f a u ar + . . . a^" 1 have factors in common. 

 "We have the two equations, 



PQ^=QP'-PQ' + P W ', i 



hence, since 7— is a fraction in the lowest terms, any common factors of 



ty^x and a 0 +a L x+ . . . -f-a w ,r w must be factors of P or Q ; hence if x— a be 

 one of the factors of a 0 + aj#+ ... + cc, n x m , we may ascertain if it is a 

 factor of P and Q by putting in the proposed differential equation 



y=A m (x-a) m + A m+1 (x-a)™+i+A m+2 (x-a) m + 2 + 

 and shall thus obtain an equation to determine the index (m) ; and we must 

 treat the other factors of a 0 + a 1 -r + a / r+ . . . a m x m in the same way, and 

 thus ascertain those which are also factors of P and Q. 



I shall illustrate these remarks by applying them to the well-known dif- 

 ferential equation 



M T e have 



d 2 u 2 n 



— -, — z — - u — q~u = 0. 



dx~ x- 



jU? — l(t+ ])u — q 2 x 2 u = 0. 

 dx" 



Let 



Substituting 1)— t(i+l)sa0, whence u=—ij putting then u— ^7 



d*2 



dx 2 dx 



P 



hence z=E(.r) e**, if the equation can be integrated in the form j/=-^e w J 

 which gives us 



Putting 



E(o?) — ff 0 + ff 1 ar + « 2 a' 2 4- • • • , 



we have 



m (m — 2t — 1) a m + 2q(m — i—l )a m - 1 = 0, 



