Let $\psi,\varphi\in L^2(-1,1),$ and $\lambda,\chi$ be constants to be determined.
First we got a integral equation $$\int_{-1}^1\frac{\sin c(x-y)}{\pi(x-y)}\psi(y)dy=\lambda\psi(x),\qquad|x|\le1.\tag{1}$$
Next, we got a differential equation $$\frac{d}{dx}(1-x^2)\frac{d\varphi}{dx}+(\chi-c^2x^2)\varphi=0.\tag{2}$$
For every real $x$, we further define an operator $P_x$ by $$P_x=\frac{d}{dx}(1-x^2)\frac{d}{dx}-c^2x^2.$$
The author claimed that, the operator $P_x$ commutes with the kernel of the integral operator above, which means, $$P_x\int_{-1}^1\frac{\sin c(x-y)}{\pi(x-y)}r(y)dy=\int_{-1}^1\frac{\sin c(x-y)}{\pi(x-y)}P_yr(y)dy$$ for all $r(y)\in L^2(-1,1)$ and all real $x$.
My first problem is how to prove that they commute with each other.
And then the author claimed that, every solution to $(1)$ is also a solution to $(2)$, I can not prove this too.
My second problem is how to prove that every solution to $(1)$ is also a solution to $(2)$.
My attempt for the second problem: I tried to plug $(1)$ into the left hand side of $(2)$, and to see if it was equal to $0$. But after plugging it in, many quotients involving $sin$ and $cos$ came out, which make the terms very ugly and let me hard to go on.
Any help will be appreciated.