We consider the differential equation $$\left(E_1\right): \quad y' + 0.48y = \frac{1}{250}$$ where $y$ is a function of the variable $t$ belonging to the interval $[0; +\infty[$.
We consider the constant function $h$ defined on the interval $[0; +\infty[$ by $h(t) = \frac{1}{120}$. Show that the function $h$ is a solution of the differential equation $(E_1)$.
Give the general form of the solutions of the differential equation $y' + 0.48y = 0$.
Deduce the set of solutions of the differential equation $(E_1)$.
Part B
We are now interested in the evolution of a population of bacteria in a culture medium. At an instant $t = 0$, an initial population of 30000 bacteria is introduced into the medium. We denote $p(t)$ the quantity of bacteria, expressed in thousands of individuals, present in the medium after a time $t$, expressed in hours. We therefore have $p(0) = 30$. We admit that the function $p$ defined on the interval $[0; +\infty[$ is differentiable, strictly positive on this interval and that it is a solution of the differential equation $(E_2)$: $$p' = \frac{1}{250} p \times (120 - p)$$ Let $y$ be the function strictly positive on the interval $[0; +\infty[$ such that, for all $t$ belonging to the interval $[0; +\infty[$, we have $p(t) = \frac{1}{y(t)}$.
Show that if $p$ is a solution of the differential equation $(E_2)$, then $y$ is a solution of the differential equation $\left(E_1\right): \quad y' + 0.48y = \frac{1}{250}$.
We admit conversely that, if $y$ is a strictly positive solution of the differential equation $(E_1)$, then $p = \frac{1}{y}$ is a solution of the differential equation $(E_2)$. Show that, for all $t$ belonging to the interval $[0; +\infty[$, we have: $$p(t) = \frac{120}{1 + K\mathrm{e}^{-0.48t}} \text{ with } K \text{ a real constant.}$$
Using the initial condition, determine the value of $K$.
Determine $\lim_{t \rightarrow +\infty} p(t)$. Give an interpretation of this in the context of the exercise.
Determine the time required for the bacterial population to exceed 60000 individuals. The result will be given in the form of a rounded value expressed in hours and minutes.
\section*{Part A}
We consider the differential equation
$$\left(E_1\right): \quad y' + 0.48y = \frac{1}{250}$$
where $y$ is a function of the variable $t$ belonging to the interval $[0; +\infty[$.
\begin{enumerate}
\item We consider the constant function $h$ defined on the interval $[0; +\infty[$ by $h(t) = \frac{1}{120}$. Show that the function $h$ is a solution of the differential equation $(E_1)$.
\item Give the general form of the solutions of the differential equation $y' + 0.48y = 0$.
\item Deduce the set of solutions of the differential equation $(E_1)$.
\end{enumerate}
\section*{Part B}
We are now interested in the evolution of a population of bacteria in a culture medium.\\
At an instant $t = 0$, an initial population of 30000 bacteria is introduced into the medium. We denote $p(t)$ the quantity of bacteria, expressed in thousands of individuals, present in the medium after a time $t$, expressed in hours.\\
We therefore have $p(0) = 30$.\\
We admit that the function $p$ defined on the interval $[0; +\infty[$ is differentiable, strictly positive on this interval and that it is a solution of the differential equation $(E_2)$:
$$p' = \frac{1}{250} p \times (120 - p)$$
Let $y$ be the function strictly positive on the interval $[0; +\infty[$ such that, for all $t$ belonging to the interval $[0; +\infty[$, we have $p(t) = \frac{1}{y(t)}$.
\begin{enumerate}
\item Show that if $p$ is a solution of the differential equation $(E_2)$, then $y$ is a solution of the differential equation $\left(E_1\right): \quad y' + 0.48y = \frac{1}{250}$.
\item We admit conversely that, if $y$ is a strictly positive solution of the differential equation $(E_1)$, then $p = \frac{1}{y}$ is a solution of the differential equation $(E_2)$.\\
Show that, for all $t$ belonging to the interval $[0; +\infty[$, we have:
$$p(t) = \frac{120}{1 + K\mathrm{e}^{-0.48t}} \text{ with } K \text{ a real constant.}$$
\item Using the initial condition, determine the value of $K$.
\item Determine $\lim_{t \rightarrow +\infty} p(t)$. Give an interpretation of this in the context of the exercise.
\item Determine the time required for the bacterial population to exceed 60000 individuals.\\
The result will be given in the form of a rounded value expressed in hours and minutes.
\end{enumerate}