Justify that there exists a unique solution $u$ to the Cauchy problem $\left( C _ { \ell } \right)$, give its expression and draw its variation table.
$$\left( C _ { \ell } \right) : \left\{ \begin{array} { l } u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) ) \\ u ( 0 ) = 0 \end{array} . \right.$$