$$
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$$
Computing with Dirichlet and Neumann conditions; details
$$ A_{i,j} = (\baspsi_j',\baspsi_i') = \int_{0}^1 \baspsi_i'(x)\baspsi_j'(x)dx
= \int_0^1 (i+1)(j+1)(1-x)^{i+j} dx,
$$
Choose \( f(x)=2 \):
$$
\begin{align*}
b_i &= (2,\baspsi_i) - (D,\baspsi_i') -C\baspsi_i(0)\\
&= \int_0^1 \left( 2(1-x)^{i+1} - D(i+1)(1-x)^i\right)dx -C\baspsi_i(0)
\end{align*}
$$
Can easily do the integrals with sympy. \( N=1 \):
$$
\begin{equation*}
\left(\begin{array}{cc}
1 & 1\\
1 & 4/3
\end{array}\right)
\left(\begin{array}{c}
c_0\\
c_1
\end{array}\right)
=
\left(\begin{array}{c}
-C+D+1\\
2/3 -C + D
\end{array}\right)
\end{equation*}
$$
$$ c_0=-C+D+2, \quad c_1=-1,$$
$$ u(x) = 1 -x^2 + D + C(x-1)\quad\hbox{(exact solution)} $$