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Application op the Cauchy Parameter Method to the Solution 

 OF Difference Equations, 



By T. E. Mason. 



In the application of tlie Caucliy parametei' metluKl to the solution 

 of difference equations the following are the necessary steps: 



1) Break the equation up into two parts, one of which gives a part 

 f)(x) which may be readily solved and multiply tlie other part of the 

 equation by the parameter t, so that the equation 



f(x)=0 

 becomes 



(a) fi(x)+tf,(x)=0. 



2) Assume a solution of the form 



U(x)=A(x)+B(x)t+C(x)tHD(x)t''+ ■ 



3) Substitute 'in the etpiation (a) and eipiate the coefficients of the 

 different powers of t to zero and solve. Then the parameter t is made 

 equal to 1. 



4) The solution 



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



must be shown to be convergent and to satisfy the original equation. 



In breaking up the equation it is necessary to make such division 

 that the resulting solution is convergent. In equations with constant 

 coefficients the solution of the resulting equations is, in general, no easier 

 than the solution of the original equation, so that this methotl of solution 

 ib of little or no value there. 



By a proper division of the equation the method of Cauchy will give 

 the same results as the method of successive approximations. Let us illus- 

 trate this by means of the example 



AU(x)=<l'(x)U(x), 

 where* 



*(x)=*"x ' + *"'x^+ 



*The general linear homogeneous difference equation of first order may be transformed to this 

 form by a transformation of the form 



g(x)=xa^a^x'"f(x), 



where the a, a and m are constants to be determined for the particular equation. 



