Context
Part A
Generally, I like Handbook on Differential Equations by Zwillinger [1]. It seems to meet my needs most of the time. For example, from this book I was able to learn how to solve linear nonhomogeneous differential equation subject to homogeneous boundary conditions. However, I can't seem to get my head around their explanation or their example regarding Green's function solutions to linear homogeneous differential equation subject to nonhomogeneous boundary conditions.
Part B
I have looked through this site for exemplary questions on this theme. I have found [2], [3], and [4] are related to this question to some extent. In [2], the question indeed revolves around the a homogeneous differential equation with nonhomogeneous Dirichlet boundary conditions. In fact, the differential operator that I use in this question is exactly the same as that appearing in [2]. There is an accepted answer in [2]. However, the answer there does not suit my needs. I cannot figure out what is being done, how come, and I can not generalize the results to more complex problems. In [3], the OP asks about the Cauchy problem. Again there is an answer but I am not able to generalize that answer to solve additional problems. Thus, in my question below, I am asking for answers that adhere closely to what is found in [1] (see below). In [4], we have the same linear operator, but with mixed boundary conditions. There is an answer in [4], except that the answer includes the Green's function $G$ (i.e. a Green's function for the linear nonhomogeneous differential equation subject to homogeneous boundary conditions) rather than $g$ (i.e. a Green's function for the homogeneous differential equation subject to nonhomogeneous boundary conditions (see below)).
Part C
Here is what Zwillinger has to say on the matter. Please note that under fair use, I am copying verbatim from [1]:
``... suppose we want to solve the linear homogeneous differential equation subject to inhomogeneous boundary conditions \begin{align} L\left[v\right] &= 0, \\ B [v] &= h(\mathbf{x}),\quad \mathbf{x}\in \partial \Omega. \end{align} If we can solve \begin{align} L\left[g(\mathbf{x};\mathbf{z})\right] &= 0, \\ B [g(\mathbf{x};\mathbf{z})] &= \delta(\mathbf{x}-\mathbf{z}), \end{align} for $g(\mathbf{x},\mathbf{z})$, then the solution to the boundary value problem is given by $$ v(\mathbf{x}) =\int_\Omega g(\mathbf{x};\mathbf{z})h( \mathbf{z}) d\mathbf{z}. $$ ... The conditions on $g(\mathbf{x};\mathbf{z})$ are ... (a) $L[g(\mathbf{x};\mathbf{z}) ] = 0$. (b) $B[g(\mathbf{x};\mathbf{z}) ] = 0$, except at $\mathbf{x}=\mathbf{z}$. (c) If $L[\cdot]$ is an $n$th order differential equation, then $g(\mathbf{x};\mathbf{z})$ must be continuous (with its derivatives up to order $n-2$) at $\mathbf{x}=\mathbf{z}$. (d) $\int_{\mathbf{z}^-}^{\mathbf{z}^+} B[g(\mathbf{x};\mathbf{z})]d\mathbf{x}=1.$
Questions
Q.1
How can I construct a Green's function, $g$, and use it solve the linear homogeneous differential equation for $\Phi(x)$ for $x \in [a,b]\subset\mathbb{R}$, \begin{align} \frac{d\Phi^2}{dx^2} &= 0, \end{align} subject to inhomogeneous boundary conditions \begin{align} &&B [\Phi] = \int_{\mathbb{R}} V_a\delta(x-a) + V_b\delta(x-b) dx ? \end{align}
Q.2
How can I show that the Green's function, $g$, satisfies the four conditions given in the block quote in the context?
Q.3
If there is, in fact, an error in the language in the block quote to make it correct, then can you explain what it it?
Bibliography
[1] Zwillinger and Dobrushkin, Handbook on Differential Equations, Fourth Edition, pp. 209-18.
[2] Confusion about Green's functions for inhomogeneous boundary conditions but no forcing.
[3] Using Greens function to solve homogenous wave equation with inhomogeneous boundary conditions
