Working through "Linear Partial Differential Equations and Fourier Theory" by Marcus Pivato I’ve come across exercise 14G.1 which is to prove the following:
‘For any fixed $\text{n}\in\Bbb{N}$ and $\lambda>0$, the Bessel equation \begin{equation} r^2 \frac{d^2}{dr^2}R(r)+r\frac{d}{dr}R(r)+(\lambda^2r^2-n^2)R(r)=0 \end{equation} has the solution $R(r):=J_n(\lambda r)$.’
The n-th order Bessel function has already been given by its power series as a solution to the n-th order Bessel equation. Information on the derivatives of $J_n(x)$ are derived at a later point in the text, so my assumption is that any recurrence relations between $J_n^\prime$ and $J_n$ aren’t required to solve this problem.
I’m struggling to solve this particular task, even after multiple different approaches. Any help and/or nudges in the right direction would be greatly appreciated.