277 



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

 powers of t, we have 



A2A(x)-x'"' = or d2A(x)=x<°' 

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

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

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



A2A(x)=x(''' 



xCn+l) 



A A(x) = +pi(x) 



n + 1 



x(n+2) 



-•^^^^-7 Tn^+P^'^^^ .X + p,(x) 



(n+2)(-> 



ax(n+2) 



A--^B(x)=- -^ + api(x) . x + ap2(x) 



(n-|-2)<2) 



„2x(n+4) ^(3) y(2) 



A-C;x) = ; -^+a-pi(x) + a=P2vx) 



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



a2x(n+6) x^^^ x*"** 



C(x) = — - + a=pi(x) — + a=p2(x) — 



(n+6)(6) 5! 4! 



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



.-<2; n2v(4) „3v(6) 



U(x)=^ — + - ~-^~ — - + +pi(x) x + + — + 



' (n+6)**^) L 3! 5! 



1 . , r, ax<2^ a^x^*' a^x^S) l 



. +p.(xj 1 + + + + 



J L 2 4 6! J 



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

 tution that this does afford a solution of the equation. 



If we denote the solution of the previous equation by U*°*(x), 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 + ayi) + a2X<2) + a3x(3)+ +anix('"\ 



may be written in the form 



m 



U(x) = 2 a„U'°Kx). 

 n = 



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



The solution of other examples would follow the same method. 

 Bloominijtoti, Incl. 



