This is part of a textbook on differential equation. The author intends to express the Bessel equation as a Sturm Liouville equation.
Bessel’s equation as a Sturm–Liouville equation. The Bessel function $J_{n}(x)$ with fixed integer $n\geq 0$ satisfies Bessel’s equation $x^{2}y''+xy'+(x^{2}-n^{2})y=0$ so we can write $${\tilde x}^{2}\ddot{J_{n}}(\tilde x)+\tilde x{\dot J}_{n}(\tilde x)+({\tilde x}^{2}-n^{2})J_{n}(\tilde x)=0$$ where ${\dot J}_{n}={{dJ_{n}}\over{d\tilde x}}$ and ${\ddot J}_{n}={{d^{2}J_{n}}\over{d{\tilde x}^{2}}}$. we set $\tilde x=kx$ then $x={{\tilde x}\over{k}}$ and by the chain rule, ${\dot J}_{n}={{dJ_{n}}\over{d\tilde x}}={{dJ_{n}}\over{dx}}/k={{{J_{n}}'}\over{k}}$ and ${\ddot J}_{n}={{{J_{n}}''}\over{k^{2}}}$. In the first two terms of Bessel’s equation, $k^{2}$and $k$ drop out and we obtain $$x^{2}{J_{n}}''(kx)+x{J_{n}}'(kx)+(k^{2}x^{2}-n^{2})J_{n}(kx)=0$$ Dividing by $x$ and using ${\left({x{J_{n}}'(kx)}\right)}'=x{J_{n}}''(kx)+{J_{n}}'(kx)$ gives the Sturm–Liouville equation: $${\left[{x{J_{n}}'(kx)}\right]}'+\left({-{{n^{2}}\over{x}}+\lambda x}\right)J_{n}(kx)=0$$ where $\lambda=k^{2}$. this is a Sturm–Liouville equation $${\left[{p(x)y'}\right]}'+\left({q(x)+\lambda r(x)}\right)y=0$$ with $p(x)=x$, $q(x)=-{{n^{2}}\over{x}}$, $r(x)=x$ and parameter $\lambda=k^{2}$.
[advanced engineering mathematics 10th ed by Erwin Kreyszig] I think there is a problem with relation
$$x^{2}{J_{n}}''(kx)+x{J_{n}}'(kx)+(k^{2}x^{2}-n^{2})J_{n}(kx)=0$$
In fact, according to the chain rule
$${\dot J}_{n}(\tilde x)={{dJ_{n}(\tilde x)}\over{d\tilde x}}={{dJ_{n}(\tilde x)}\over{dx}}{{dx}\over{d\tilde x}}={{dJ_{n}(kx)}\over{dx}}\cdot{{1}\over{k}}={{{\left[{J_{n}(kx)}\right]}'}\over{k}}$$
And in a similar way
$${\ddot J}_{n}(\tilde x)={{{\left[{J_{n}(kx)}\right]}''}\over{k^{2}}}$$
Therefore, by inserting these relations into Equation
$${\tilde x}^{2}\ddot{J_{n}}(\tilde x)+\tilde x{\dot J}_{n}(\tilde x)+({\tilde x}^{2}-n^{2})J_{n}(\tilde x)=0$$
we get
$$x^{2}{\left[{J_{n}(kx)}\right]}''+x{\left[{J_{n}(kx)}\right]}'+(k^{2}x^{2}-n^{2})J_{n}(kx)=0$$
relation ${\left({x{J_{n}}'(kx)}\right)}'=x{J_{n}}''(kx)+{J_{n}}'(kx)$ is also wrong. The correct form of this equation is $${\left[{x{\left[{J_{n}(kx)}\right]}'}\right]}'=x{\left[{J_{n}(kx)}\right]}''+{\left[{J_{n}(kx)}\right]}'$$
Using this relation and dividing by $x$, we get
$${\left[{x{\left[{J_{n}(kx)}\right]}'}\right]}'+\left({-{{n^{2}}\over{x}}+\lambda x}\right)J_{n}(kx)=0$$
In summary, in my opinion, ${\left[{J_{n}(kx)}\right]}'$ should be used instead of ${J_{n}}'(kx)$ Because these two expressions are not the same. indeed ${\left[{J_{n}(kx)}\right]}'=k{J_{n}}'(kx)$.
This image shows the section from the book along with the corrections I made
What you think? Am I right or is the book written correctly?
I think the author was basically correct. For typesetting convenience let $u= \overline x$. Consider the author's change of variables in which $u= kx$. Then $ u^2\frac{d^2}{du^2} = k^2 \frac{d^2}{k^2 d^2 x}$. Note that the powers of $k$ cancel. Ditto for $ u\frac{d}{du}$. So the Bessel equation looks essentially the same in either set of letters. The only exception to such cancellation of powers is the undifferentiated last term $ (u^2- n^2) J(u)$ which becomes $(k^2 x^2 -n^2) J_n(kx)$. (This is the only term in the Bessel equation that acquires an occurrence of $k^2$ under this substitution.)
To rearrange the Bessel equation to the Sturm-Liouville format it might have been cleaner if the author had done that step first, using only the $u$ variable. Start with the Bessel equation $ u^2 J''(u) + \ldots$; then divide by $u$; then write $(uJ'(u))'= uJ''+ J'$ etc. At the very end replace $u=kx$.