27G r I 



Let us write 



AU(x)-t0(x)U(x)=n 

 and assume for the solution 



U(x) = A(x) +B(x)t + C(x)t = + D(x)t3+ 



Then 



AU(x) = AA(x) +:iB(x)t + ^C(x)t= + AD(x)t3+ .... 

 Substituting in the equation we have 



AA(x) +t[AB(x) -ci(x)A(x)] + tnAC(x) -^(x)B(x)] + tnAD(x) -9(x)C(x)]+ . . =0. 



Equating 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) =^(x)B(x) 



AD(x)-9(x)C(x)=0 or AD(x) =^(x)C(x) 



Solving we have 



A(x) = 1 



B(x)^Sx^(x), where Sx<?(x) = - 2 ^(x + i) 



i=0 

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



D(x)=Sx0(x)Sx0(x)Sx^(x) 



.•.U(x) = 1 + Sx9(x) +Sx^(x)Sx<^(x) +Sx«/>(x)Sx^(x)Sx0(x) + 



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

 solution 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 difTcrent from those obtained by the ordinary method of successive 



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



equation 



A2U(x)-aU(x)=x'"»t, 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 Mathemathical Society, Vol. 12, No. 1, p. 101. If in that 

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

 tx(°)=x(x-l) (x-2^ (x-n+1). 



