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    $\begingroup$ Well, this type of problem is just… hard. Have you tried using solution for small $\lambda$ (or small rEnd1) as initial guess and increase its value slowly? Here's an example. $\endgroup$ Commented May 20, 2018 at 3:45
  • $\begingroup$ @xzczd thanks for the comment. I did start with small $\lambda$ but due to the demanding precision of the initial guess, I don't think the solution initial value for small $\lambda$ can be used for large $\lambda$ case. As for lower rEnd1, I tried that too. The problem is that lowering rEnd1 requires a good knowledge of the asymptotic behavior of $\Phi$ at large r but I couldn't find one. If I lower rEnd1 and demand $\Phi(rEnd1) =0$ with brute-force, the shape of the solution is changed a lot, unfortunately. $\endgroup$ Commented May 20, 2018 at 4:27
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    $\begingroup$ Why are you assuming A[rStart1] == 1 in the λ == 300 computation? $\endgroup$ Commented May 20, 2018 at 13:24
  • $\begingroup$ @bbgodfrey that are two evidences. 1) small $r$ expansion, 2) numerically, if I set A(0) equal to a different value, say 1.3, it will be pulled down to 1 immediately. I have added some code for the expansion. $\endgroup$ Commented May 20, 2018 at 14:46
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    $\begingroup$ There are some useful tools for formatting posts here and here. You may also find this helpful. $\endgroup$ Commented May 20, 2018 at 16:06