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## PDE, inhomogeneous diffusion equation

 Quote by Chestermiller The solution is expressible as Bessel functions. See Eqn. 9.1.52 in Abramowitz and Stegan.
Thanks a lot for the reference (it's a free ebook).
So it states that the solution to the DE $w''-\frac{2\nu -1 }{z}w'+\lambda ^2 w =0$ is $w =z^{\nu } C_{\nu} (\lambda z )$. Where C could be any linear combination of the Bessel functions of any kind.
My problem is... if I compare the DE to solve (2) or a slightly simplified but equivalent to DE: (3): $R''+ \frac{2}{r}R'+R(\lambda - m )=0$, one finds that $\nu =-\frac{1}{2}$. So clearly nu isn't the smaller roots of the indicial equation as I thought.
What is very strange is that Wolfram alpha gives a totally different answer which seems to me unrelated to the Bessel function (http://www.wolframalpha.com/input/?i...y%27%2By*a%3D0) and it seems to confirm that $c=-1$ is the smaller root of the indicial equation.
According to the solution in the book of Abramowitz and Stegun the solution would be of the form of $\frac{C_{-\frac{1}{2}}(r \sqrt {\lambda -m})}{\sqrt r}$.
I'm totally confused.

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 Quote by fluidistic Thanks a lot for the reference (it's a free ebook). So it states that the solution to the DE $w''-\frac{2\nu -1 }{z}w'+\lambda ^2 w =0$ is $w =z^{\nu } C_{\nu} (\lambda z )$. Where C could be any linear combination of the Bessel functions of any kind. My problem is... if I compare the DE to solve (2) or a slightly simplified but equivalent to DE: (3): $R''+ \frac{2}{r}R'+R(\lambda - m )=0$, one finds that $\nu =-\frac{1}{2}$. So clearly nu isn't the smaller roots of the indicial equation as I thought. What is very strange is that Wolfram alpha gives a totally different answer which seems to me unrelated to the Bessel function (http://www.wolframalpha.com/input/?i...y%27%2By*a%3D0) and it seems to confirm that $c=-1$ is the smaller root of the indicial equation. According to the solution in the book of Abramowitz and Stegun the solution would be of the form of $\frac{C_{-\frac{1}{2}}(r \sqrt {\lambda -m})}{\sqrt r}$. I'm totally confused.
For half integer orders the Bessel functions are basically just combinations of signs and cosines divided by some power of x.

See http://en.wikipedia.org/wiki/Spheric...ions:_jn.2C_yn

You should be able to mash the solution from Abramowitz and Stegun into looking like the one wolfram alpha gave you.

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