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I am trying to solve an ODE by casting it as an operator problem, say ##K[y(x)] = \lambda M[y(x)]##, where ##y## is a trial function, ##x## is the independent variable, ##\lambda## is the eigenvalue, and ##K,M## are linear differential operators. For this particular problem, it's easier for me to work with the inverse operator problem ##M^{-1}[y(x)] = \lambda K^{-1}[y(x)]##. Constructing inverse operators implies building a Green's function.

The technique I've used to build a Green's function is variation of parameters, which takes the form ##G = y_1(x)y_2(\xi) / w_\alpha## where ##y_1,y_2## are fundamental solutions associated with a particular operator, say ##K##, and ##w_\alpha## is their associated Wronskian, where the subscript ##\alpha## is a parameter. I observe ##\alpha \to 0 \implies w\to 0##. Does this imply any ##\alpha \neq 0## yields the correct Green's function? What if ##\alpha## is very VERY small but not zero? Is this something that can cause numerical issues?

I ask this because an analytic solution for the operator ODE exists for the ##\alpha = 0## case, but this causes issues with the Wronskian. When benchmarking, how ``small'' of an ##\alpha## should I use? I can provide more information if someone is willing to help and needs more understanding, as I have not really mentioned specifics regarding the numerics.