$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Example with two Dirichlet values; details
Insert \( u = B + \sum_j c_j\baspsi_j \) in variational formulation:
$$ (u',v') = (2,v)\quad\Rightarrow\quad (\sum_jc_j\baspsi_j',\baspsi_i')
= (2-B',\baspsi_i)\quad \forall v\in V$$
$$
\begin{align*}
A_{i-1,j-1} &= \int_0^L \basphi_i'(x)\basphi_j'(x) \dx\\
b_{i-1} &= \int_0^L (f(x)\basphi_i(x) -
B'(x)\basphi_i'(x))\dx,\quad B'(x)=C\basphi_{0}'(x) + D\basphi_{N_n}'(x)
\end{align*}
$$
for \( i,j = 1,\ldots,N+1=N_n-1 \).
New boundary terms from \( -\int B'\basphi_i'\dx \): \( C/2 \) for \( i=1 \)
and \( -D/2 \) for \( i=N_n-1 \)
