277 



Substituting in the equation and equating to zero the coefficients of the 

 powers of t, we have 



A=A(x)-x'°' = or A2A(x)=x(°' 

 A2B(x)-aA(x-)=0 or A=B(x)=aA(x) 

 A2C(x)-aB(x)=0 or A=C(x) = aB(x) 

 A2D(x)-aC(x)=0 or A-D(x) = aC(x) 



A=A(x)=x<°) 



AA(x) = ^^ +pi(x) 



n + 1 



x(n+2) 



A(x)-- ^ +p/x) .x + p,(x) 



ax(n+2) 



A=B(x) = + api(x) . X + ap2(x) 



(n + 2)'2) 



„2y(n+4) „(3) „(2) 



A=C( x) = + a=pi(x) — + a=P2vx) — 



(n + 4)(''' 3! 2! 



Cvx) = —- + a=pi(x) — + a-p2(x) — - 



(n+6)'6) »- 5! 41 



. ,., . x("+2) ax(°+4) a^x(°+6) , , r ax(3) a^x^^^ a'x^^) 



1 +p/x) F] 



ax<2; a^x^'" a'x^^^ 



Since a<l these series converge, and it can readily be shown by substi- 

 tution that this does afiford a solution of the equation. 



If we denote the solution of the previous equation by U^°Kx), then the 

 solution of the equation 



A=U(x)-aU(x) = P(x), a<l, 

 where P(x) is a polynomial in x of the form 



P(x) = ao + aix^i' + a2X<2) + asx'^' + + amx'""', 



may be written in the form 



m 



U(x) = i- anU<°'(x). 

 n = 



The 2m + 2 periodic functions combine into 2 independent ones. 



The solution of other examples would follow the same method. 

 Blornningtoii, Ind. 



