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I'm trying to do the part of this qusetion where we need to find two linearly independent solutions to (2) of the given form. Is there a nicer way to do it other than just plugging it into (2). I was thinking of trying to pick B=0 for one solution, and A=0 for another, and see if it satisfies it (since then they would be linearly independent), but that doesn't seem to work because I need two constants to match the left hand side of (2) to the right hand side of (2). Is there a nice way to do it that I am missing. Thanks!

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lkjhgfdsa
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    One way to proceed is by 'guessing' that the solution should be of the form $\psi(r)=\phi(r)e^{-\lambda r}$, then plugging this form in and showing that $\phi(r)$ will have to satisfy a simple 2nd order ODE. This is similar to the approach for the quantum harmonic oscillator, wherein one obtains Hermite's ODE from the Schrodinger equation by an appropriate ansatz. – Semiclassical Jun 12 '16 at 16:11
  • @Semiclassical Thanks for your reply! I can understand that would be the general approach, but here aren't we already given the form to be $(A+Br)e^(-λr)$ for two linearly independent solutions (the ground and first state presumably), so can't we just use that? I can see we can just plug it into the equation (2), and match the constants, but I've tried it and it gets very ugly, so was hoping to know if there was an easier way for this specific case to find two linearly independent solutions. Thanks! – lkjhgfdsa Jun 12 '16 at 16:16

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