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I'm reading the paper:

http://arxiv.org/abs/hep-ph/9907218

They have an ODE (eqn 7):

[tex]-\frac{1}{r^2}\frac{d}{d\phi}e^{-4kr\phi}\frac{dy_n}{d\phi}+m^2e^{-4kr\phi}y_n=m^2_ne^{-2kr\phi}y_n[/tex]

They then make a change of variables:

[tex]z_n=\frac{m_n}{k}e^{kr\phi}[/tex]

[tex]f_n=e^{-2kr\phi}y_n[/tex]

Then the ODE becomes:

[tex]z_n^2\frac{d^2f_n}{dz_n^2}+z_n\frac{df_n}{dz_n}+(z_n^2-[4+\frac{m^2}{k^2}])f_n=0[/tex]

My question is regarding this change of variables:

How do you 'know' how to change the variables so that the ODE comes out as this tiday Bessel function. Is this anartalmost, or is there some kind of technique??

Looking foward to gaining some insight here.

Richard

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# ODE change of vars

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