\section*{EXERCISE - B}
\textbf{Main topics covered: Function study, exponential function; Differential equations}

\section*{Part I}
Let us consider the differential equation
$$y' = -0.4y + 0.4$$
where $y$ denotes a function of the variable $t$, defined and differentiable on $[0; +\infty[$.

\begin{enumerate}
  \item a. Determine a particular constant solution of this differential equation.\\
b. Deduce the set of solutions of this differential equation.\\
c. Determine the function $g$, solution of this differential equation, which satisfies $g(0) = 10$.
\end{enumerate}

\section*{Part II}
Let $p$ be the function defined and differentiable on the interval $[0; +\infty[$ by
$$p(t) = \frac{1}{g(t)} = \frac{1}{1 + 9\mathrm{e}^{-0.4t}}$$

\begin{enumerate}
  \item Determine the limit of $p$ at $+\infty$.
  \item Show that $p'(t) = \frac{3.6\mathrm{e}^{-0.4t}}{\left(1 + 9\mathrm{e}^{-0.4t}\right)^{2}}$ for all $t \in [0; +\infty[$.
  \item a. Show that the equation $p(t) = \frac{1}{2}$ has a unique solution $\alpha$ on $[0; +\infty[$.\\
b. Determine an approximate value of $\alpha$ to $10^{-1}$ near using a calculator.
\end{enumerate}

\section*{Part III}
\begin{enumerate}
  \item $p$ denotes the function from Part II.\\
Verify that $p$ is a solution of the differential equation $y' = 0.4y(1 - y)$ with the initial condition $y(0) = \frac{1}{10}$ where $y$ denotes a function defined and differentiable on $[0; +\infty[$.
  \item In a developing country, in the year 2020, 10\% of schools have access to the internet.\\
A voluntary equipment policy is implemented and we are interested in the evolution of the proportion of schools with access to the internet.\\
We denote $t$ the time elapsed, expressed in years, since the year 2020.\\
The proportion of schools with access to the internet at time $t$ is modelled by $p(t)$.\\
Interpret in this context the limit from question II 1 then the approximate value of $\alpha$ from question II 3. b. as well as the value $p(0)$.
\end{enumerate}