Show that there exists a unique constant solution of equation $\left( E _ { \ell } \right)$, denoted $\gamma \in \mathbf { R }$, and verify that the solution $u$ found in question 1 satisfies
$$\lim _ { x \rightarrow + \infty } u ( x ) = \gamma .$$
where $\left( E _ { \ell } \right) : \quad u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) )$.