$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Example: linear approximation; solution
$$
\begin{align*}
A_{0,0} &= (\baspsi_0,\baspsi_0) = \int_1^21\cdot 1\, dx = 1\\
A_{0,1} &= (\baspsi_0,\baspsi_1) = \int_1^2 1\cdot x\, dx = 3/2\\
A_{1,0} &= A_{0,1} = 3/2\\
A_{1,1} &= (\baspsi_1,\baspsi_1) = \int_1^2 x\cdot x\,dx = 7/3\\
b_1 &= (f,\baspsi_0) = \int_1^2 (10(x-1)^2 - 1)\cdot 1 \, dx = 7/3\\
b_2 &= (f,\baspsi_1) = \int_1^2 (10(x-1)^2 - 1)\cdot x\, dx = 13/3
\end{align*}
$$
Solution of 2x2 linear system:
$$ c_0 = -38/3,\quad c_1 = 10,\quad u(x) = 10x - \frac{38}{3} $$
