8.7.3 Example C For the equation Ay(x) = Dy(x), (8.282) it follows that C y(x + 1) – y(x) = dx dy(x) (8.283) This is a linear equation, with constant coefficients, and we will assume the solution...

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Extracted text: 8.7.3 Example C For the equation Ay(x) = Dy(x), (8.282) it follows that C y(x + 1) – y(x) = dx dy(x) (8.283) This is a linear equation, with constant coefficients, and we will assume the solution takes the form y(x) = e"ª, (8.284) and, therefore, we find dy(x) y(x + 1) = e"(2+1) = e"e"ª, re"*. (8.285) dx Substitution of these results into the original equation and canceling the com- mon factor e"", gives the transcendental equation e" - r – 1 = 0. (8.286) 404 Difference Equations Note that this equation has solutions in terms of the Lambert W-function. In general, an infinite set of roots exist and if they are denoted by {rm}mE, |m=∞ then the solution takes the form Cy(2) = E ame"mz, (8.287) rmx m=0 where {am} are constants. Inspection of equation (8.286) shows that r a solution, thus one solution to Ay(x) = Dy(x) is y(x) = A, where A is an arbitrary constant. = 0 is