27C 



Let us write 



ZiU(x)-t0(x)U(x)=O 

 and assume for the solution 



U(x) = A(x) +B(x-)t + C(x)t2 + D(x)tH 



Then 



AU(x) = AA(x) +AB(x)t + AC(x)t= + AD(x)t3+ .... 



Substituting in the equation we have 



AA(x)+t[AB(x)-0(x)A(x)] + tnAC(x)-0(x)B(x)] + tM^D(x)-9(x)C(x)] + ..=O. 



Equatinf^ the coefficients of the powers of t to zero we have 

 AA(x)=0 

 AB(x)-^(x)A(x)=0 or AB(x) =?.(x)A(x) 

 AC(x)-^(x)B(x)=0 or AC(x)=0(x)B(x) 

 AD(x)-?.(x)C(x)=0 or AD(x)=0(x)C(x) 



Solving we have 



A(x) = l 



B(x)=Sx^Kx), where Sx^(x) = - ^ ^(x + i) 



i = 



C(x)=Sx0(x)Sx^(x) 



D(x)=Sx^(x)Sx^(x)Sx^'.(x) 



.•.U(x) = 1 + Sx0(x) +Sx0(x)Sx<?(x) +S.v^(x)S.x^(x)Sx0(x) + 



This series has been proven to be convergent* and gives a particuhir 

 S()luth)n of the linear homogeneous e(iuation of the first order. 



But this parameter method may be applied in such a way as to obtain 



solutions different from those obtained by the ordinary method of successive 



approximations. We shall illustrate this remark by the solution of the 



equation 



A=U(x)-aU(x)=x<"it a<l. 



Let us write 



A2U(x)-x("'-taU(x)=0 

 and assume the; solution 



U(x) = A(x) +B(x)t + C(x)t» + D(x)t»+ .... 



•Carmichael, Transactions American Mathcmathical Society, Vol. 12, No. 1, p. 101. If in that 

 discussion wo put a=l, m = 0, the two problems are identical. 

 tx(")=x(x-l) (x-2^ (x-n+1). 



