Differential Operators on Bessel Functions

Question

I am currently reading through Advanced Real Analysis by Anthony Knapp, and I've been stuck on a bit of a computation that the author seems to glance over. The larger problem is proving $\int_0^1J_0(k_nr)J_0(k_mr)r dr=0$ for $n \neq m$ where $J_0$ is the solution to the Bessel Equation ($t^2D^2+tD+t^2=0$) of the first kind, and $k_{n,m}$ are zeroes of $J_0$. The book instructs the exercise to be completed by multiplying the relevant differential operators to prove self adjointness, which I can do, and then use Green's Formula to prove the given identity. The hints for the exercises gives the following statements. Let the differential operator $\mathcal{L}(u)=(tu')'+tu$. Then $\mathcal{L}(J_0(k\cdot))=-k^2t$ given $J_0(k)=0$. It is this statement that I cannot seem to reach, not matter how I manipulate the problem. It's this statement I'm stuck on. How would you arrive at that particular equality given just these conditions? The remaining statements I can work out myself where the integral is actually solved.