I am taking a first course on ordinary differential equations. I was very interested to learn more about Green's functions. According to this page, Green's functions are a device to solve difficult ordinary and partial differential equations, which may be unsolvable by other methods. Green's functions are also used widely in electrodynamics and quantum field theory.
I'm reading the book Kreider, Kuller, Ostberg and Perkins(KKOP). On page 149, a Green's function is defined as follows -
Definition. A function $H(x,t)$ is said to be a Green's function for initial value problems involving the linear differential operator L, if and only if $H(x,t)$ satisfies the properties -
(1) $H(x,t)$ is defined throughout the region $R$ of $xt$-plane consisting of all points $(x,t)$ with $x,t$ lie in the interval $I$.
(2) $H(x,t)$ and all its partial derivatives $\partial H/\partial x$, $\partial^2 H/\partial x^2$, $\ldots$, $\partial^n H/\partial x^n$ are continuous everywhere in $R$;
(3) For every $x_0$ in $I$, and every $h$ in $\mathcal{C}(i)$, the function
$$y(x)=\int_{x_0}^{x}H(x,t)h(t)dt$$
is a solution of the initial-value problem
$$Ly=h$$
$$y(x_0)=y'(x_0)=\ldots=y^{(n-1)}(x_0)=0$$
So, the Green's operator is the right inverse of a linear differential operator $L=D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$, since
$$\bbox[border:2px solid blue]{G(h)=\int_{x0}^{x}H(x,t)h(t)dt=y(x)}$$
and thus,
$\begin{align} Gh &=y\\ \therefore LGh &=Ly\\ \therefore LGh &=h \end{align}$
Further, since the function $Gh(x)$ is a solution of the initial value problem $Ly=h, Gh(x_0)=Gh'(x_0)=\ldots=Gh^{(n-1)}(x_0)=0$, every element $h$ in $\mathcal{C}(I)$ has a unique pre-image $y$ in $\mathcal{C}^{n}(I)$. The inverse operator $G$ is unique for a given linear differential operator $L$.
The book then proves the following identities that hold for Green's functions. I state them without proof for brevity.
Theorem. Let $H(x,t)$ be defined throughout the region $R$ described above and suppose that $H$ and its partial derivatives $H_1,H_2,\ldots,H_n$ are continuous everywhere in $R$. Then, $H(x,t)$ is a Green's function for the linear differential operator $L=D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$ if an only if the following identities are satisfied throughout $R$:
$$\begin{align} H(x,x) &= 0\\ H_1(x,x) &= 0 \\ H_2(x,x) &= 0 \\ \vdots\\ H_{n-1}(x,x) &= 1 \end{align}$$
and
$$H_n(x,t)+a_{n-1}(x)H_{n-1}(x,t)+\ldots+a_1(x)H_1(x,t)+a_0(x)H(x,t)=0$$
Theorem. The Green's function for a constant coefficient linear differential operator $L$ can be written in the form $k(x-t)$, where $k(x)$ is the solution on $(-\infty,\infty)$ of the initial value problem
$$Ly=0$$
$$y(0)=y'(0)=\ldots=y^{(n-2)}(0)=0, y^{(n-1)}(0)=1$$
Questions.
1) What are some interesting applications of Green's functions in solving problems in Physics?
2) Does the Green's function $k(x-t)$ for constant coefficient linear differential operators have special significance?
3) How do I use the last theorem to find the Green's function for the linear differential operator $D^2(D-1)$? I used the analytical formula to obtain the $K(x,t)=e^{(x-t)}-1-(x-t)$.