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ODE with Bessel decay

Discussion in 'Mathematics' started by AMath91, Oct 8, 2018.

  1. AMath91

    AMath91 Guest

    This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.

    I would like to estimate the asymptotic behaviour of the solution of the following ODE $$ w''(r)+\frac{1}{r}w'(r)-k^{2}w(r)=-g(r)$$ where $g$ is a smooth positive function function satisfying $$ g(r) = O\left(\frac{e^{-kr}}{\sqrt{r}}\right) \ \ \ \ \ r\to +\infty . $$

    Assuming that $w(r)$ decays exponentially fast, is it true that w(r) has exactly the same asymptotic decay as g(r)?

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