Consider the differential equation
$$\left( E_0 \right) : \quad y^{\prime} = y$$
where $y$ is a differentiable function of the real variable $x$.

\begin{enumerate}
  \item Prove that the unique constant function solution of the differential equation $\left( E_0 \right)$ is the zero function.
  \item Determine all solutions of the differential equation $(E_0)$.
\end{enumerate}

Consider the differential equation
$$(E) : \quad y^{\prime} = y - \cos(x) - 3\sin(x)$$
where $y$ is a differentiable function of the real variable $x$.

\begin{enumerate}
  \setcounter{enumi}{2}
  \item The function $h$ is defined on $\mathbb{R}$ by $h(x) = 2\cos(x) + \sin(x)$. It is admitted that it is differentiable on $\mathbb{R}$. Prove that the function $h$ is a solution of the differential equation $(E)$.
  \item Consider a function $f$ defined and differentiable on $\mathbb{R}$. Prove that: ``$f$ is a solution of $(E)$'' is equivalent to ``$f - h$ is a solution of $\left(E_0\right)$''.
  \item Deduce all solutions of the differential equation $(E)$.
  \item Determine the unique solution $g$ of the differential equation $(E)$ such that $g(0) = 0$.
  \item Calculate:
$$\int_{0}^{\frac{\pi}{2}} \left[ -2\mathrm{e}^{x} + \sin(x) + 2\cos(x) \right] \mathrm{d}x$$
\end{enumerate}