I've been asked to find the bounded eigenvalues and eigenfunctions of this DE in this form:
$$\frac{d}{dx}\left(x\frac{dy}{dx}\right) = \lambda xy$$
where $x\in[0,1]$ and $y(1)=0$
The hint given is to change the variables in the form $r=\beta x$, but I'm still really struggling to see where to go after this. I'm pretty sure that the answers are supposed to be Bessel functions (from the phrasing of the subsequent parts of the question) but I can't show it.
The Bessel functions $J_0(t)$ and $Y_0(t)$ of order $0$ satisfy the Bessel differential equation of order $0$:
$$ t^2 y'' + t y' + t^2 y = 0 $$ i.e.
$$ (t y')' = - t y$$
Thus $y(x) = J_0(\beta x)$ and $Y_0(\beta x)$ would satisfy
$$ (x y')' = - x \beta^2 y $$
Thus you want $\lambda = -\beta^2$. Since you want $y$ to be bounded at $0$, you can't have $Y_0$, so $y(x) = J_0(\beta x)$. Now you want $y(1) = J_0(\beta) = 0$, so $\beta$ must be one of the zeros of $J_0$.