Deduce that there exists a solution $u \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$ to problem (1):
$$\left\{ \begin{array} { l }
- u ^ { \prime \prime } ( x ) + c ( x ) u ( x ) = f ( x ) , x \in [ 0,1 ] \\
u ( 0 ) = u ( 1 ) = 0
\end{array} \right.$$
Show that this solution is unique.