The Question:
Consider the system
\begin{align} & Ly(x) \equiv y''(x)+y(x)=f(x),\qquad 0<x<1 \\ & y(0)=1, \qquad y(1)=0 \end{align}
(i) Find the eigenvalues and eigenfunctions of $L$
(ii) Find the eigenvalues of $M$, where
$$My(x) \equiv \int_0^1g(x,s)y(s)ds$$
and $g(x,s)$ is the associated Green's Function of the above system.
[Hint: There is no need to compute $g(x,s)$. Simply apply across the equation]
My Attempt:
(i) I found the eigenvalues $\lambda_n = 1-n^2 \pi ^2$ with corresponding eigenfunctions $y_n(x) = A_n \sin(n\pi x)$
(ii) I am not exactly sure what the hint means. I tried considering $LMy(x)$, which gives
\begin{align} LMy(x) & =L\biggl (\int_0^1g(x,s)y(s)ds \biggl) \\ & = \frac{d^2}{dx^2}\biggl (\int_0^1g(x,s)y(s)ds \biggl)+\biggl (\int_0^1g(x,s)y(s)ds \biggl) \\ & = \int_0^1\biggl( \frac{\partial^2}{\partial x^2}g(x,s)+g(x,s) \biggl)y(s)ds \end{align}
and I still don't know how to proceed.
Also, I am not sure how to use the fact that $g(x,s)$ is the associated Green's Function.
Any hints?
Here's a hint: The Green's function has the property
$$ Lg(x; s) = \delta(x-s) $$
where $\delta(x)$ is the Dirac-delta function.
Hence,
$$ L\big(My(x)\big) = \int_0^1 \delta(x-s) y(s) ds = y(x) \tag{1} $$
You'll also need to show that
$$ L\big(My(x)\big) = M\big(Ly(x)\big) = \lambda_nMy(x) \tag{2} $$
This means proving
$$ M\big(y''(x)\big) = \int_0^1 g(x;s)y''(s)ds = \int_0^1 \frac{\partial^2}{\partial s^2}g(x;s) y(s)ds $$
which can be easily done through integration by parts.