Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$ for all $i \in \{0, \ldots, n+1\}$. Show that there exists a unique family of real numbers $\left( u _ { i } \right) _ { 0 \leq i \leq n + 1 }$ satisfying
$$\left\{ \begin{array} { l }
- \frac { 1 } { h ^ { 2 } } \left( u _ { i + 1 } + u _ { i - 1 } - 2 u _ { i } \right) + c \left( x _ { i } \right) u _ { i } = f \left( x _ { i } \right) , \text { for } 1 \leq i \leq n \\
u _ { 0 } = u _ { n + 1 } = 0
\end{array} \right.$$