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Let $J_0$ denote the Bessel function of the first kind of order $\nu = 0$ (see DLMF 10.2), $$ J_0(z) = \sum_{k = 0}^\infty (-1)^k \frac{(\tfrac{1}{4} z^2)^k }{k! \Gamma(k + 1)}. $$ Put $u_0(r) = r^{1/...

In an application I encountered the ODE $$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right) \right) \left( 8\,...