The discussion centers on solving the non-constant coefficient difference equation $(x+1)y_{x+1}-(r+x)y_{x}+ry_{x-1}=0$, where r is a constant. The standard approach involves rewriting the equation to express y_{x+1} in terms of y_x and y_{x-1}, allowing for a block-wise solution. An initial value for y is necessary, typically defined over a specific range, such as 0 ≤ x ≤ 2. For instance, if y(x) is defined as x in that range, the solution for 2 ≤ x ≤ 3 can be derived as y(x) = (r+x-1)(x-1) - r(x-2)/(x). This method provides a structured way to tackle the equation based on initial conditions.
