I have the ODE $$\frac{d}{dr}\left(r\frac{dR}{dr}\right)+\left(k^2 r-\frac{m^2}{r}\right)R=0.$$ I am trying to transform this ODE into the following Bessel equation of order $m$, $$p\frac{d}{dp}\left(p\frac{dR}{dp}\right)+(p^2-m^2)R=0.$$
It was suggested to me to use the substitution $p=kr$. Then $$\frac{dR}{dr}=\frac{dR}{dp}\frac{dp}{dr}=k\frac{dR}{dp}.$$ \begin{align} \frac{d}{d\frac{p}{k}}\left(p\frac{dR}{dp}\right)+\left(kp-k\frac{m^2}{p}\right)R=0. \end{align} I do not know how to reach the required form, as I am having particular trouble altering $\frac{d}{dr}$ into $\frac{d}{dp}$.
\begin{eqnarray*} \frac{d}{dr} = \frac{dp}{dr}\frac{d}{dp}=k\frac{d}{dp} \end{eqnarray*} Now multiply this by $r$ \begin{eqnarray*} r \frac{d}{dr} =p\frac{d}{dp}. \end{eqnarray*} Multiply the original equation by $r$ & you done.