Exercise 1 Part A Consider the differential equation $$\text{(E)} \quad y' + 0{,}4y = \mathrm{e}^{-0{,}4t}$$ where $y$ is a function of the real variable $t$. We seek the set of functions defined and differentiable on $\mathbb{R}$ that are solutions of this equation.
Let $u$ be the function defined on $\mathbb{R}$ by: $u(t) = t\mathrm{e}^{-0{,}4t}$. Verify that $u$ is a solution of (E).
Let $f$ be a function defined and differentiable on $\mathbb{R}$. We denote by $g$ the function defined on $\mathbb{R}$ by: $g(t) = f(t) - u(t)$. Let (H) be the differential equation $y' + 0{,}4y = 0$.
[a.] Prove that if the function $g$ is a solution of the differential equation (H) then the function $f$ is a solution of the differential equation (E). We will admit that the converse is true.
[b.] Solve the differential equation (H).
[c.] Deduce the solutions of (E).
[d.] Determine the solution $f$ of (E) such that $f(0) = 1$.
Part B We are interested in blood glucose levels in a person who has just had a meal. The blood glucose level in $\mathrm{g.L}^{-1}$, as a function of time $t$, expressed in hours, elapsed since the end of the meal, is modelled by the function $f$ defined on $[0;6]$ by: $$f(t) = (t+1)\mathrm{e}^{-0{,}4t}$$
[a.] Show that, for all $t \in [0;6]$, $f'(t) = (-0{,}4t + 0{,}6)\mathrm{e}^{-0{,}4t}$.
[b.] Study the variations of $f$ on $[0;6]$ then draw up its variation table on this interval.
A person is hypoglycaemic when their blood glucose level is below $0{,}7\,\mathrm{g.L}^{-1}$.
[a.] Prove that on the interval $[0;6]$ the equation $f(t) = 0{,}7$ admits a unique solution which we will denote $\alpha$.
[b.] How long after having eaten does this person become hypoglycaemic? Express this time to the nearest minute.
We wish to determine the average blood glucose level in $\mathrm{g.L}^{-1}$ in this person during the six hours following the meal.
[a.] Using integration by parts, show that: $$\int_0^6 f(t)\,\mathrm{d}t = -23{,}75\,\mathrm{e}^{-2{,}4} + 8{,}75$$
[b.] Calculate the average blood glucose level in $\mathrm{g.L}^{-1}$ in this person during the six hours following the meal.
[c.] By noting that the function $f$ is a solution of the differential equation (E), explain how we could have obtained this result differently.
\textbf{Exercise 1}
\textbf{Part A}
Consider the differential equation
$$\text{(E)} \quad y' + 0{,}4y = \mathrm{e}^{-0{,}4t}$$
where $y$ is a function of the real variable $t$.\\
We seek the set of functions defined and differentiable on $\mathbb{R}$ that are solutions of this equation.
\begin{enumerate}
\item Let $u$ be the function defined on $\mathbb{R}$ by: $u(t) = t\mathrm{e}^{-0{,}4t}$.
Verify that $u$ is a solution of (E).
\item Let $f$ be a function defined and differentiable on $\mathbb{R}$.
We denote by $g$ the function defined on $\mathbb{R}$ by: $g(t) = f(t) - u(t)$.\\
Let (H) be the differential equation $y' + 0{,}4y = 0$.\\
\begin{enumerate}
\item[a.] Prove that if the function $g$ is a solution of the differential equation (H) then the function $f$ is a solution of the differential equation (E).
We will admit that the converse is true.
\item[b.] Solve the differential equation (H).
\item[c.] Deduce the solutions of (E).
\item[d.] Determine the solution $f$ of (E) such that $f(0) = 1$.
\end{enumerate}
\end{enumerate}
\textbf{Part B}
We are interested in blood glucose levels in a person who has just had a meal.\\
The blood glucose level in $\mathrm{g.L}^{-1}$, as a function of time $t$, expressed in hours, elapsed since the end of the meal, is modelled by the function $f$ defined on $[0;6]$ by:
$$f(t) = (t+1)\mathrm{e}^{-0{,}4t}$$
\begin{enumerate}
\item \begin{enumerate}
\item[a.] Show that, for all $t \in [0;6]$, $f'(t) = (-0{,}4t + 0{,}6)\mathrm{e}^{-0{,}4t}$.\\
\item[b.] Study the variations of $f$ on $[0;6]$ then draw up its variation table on this interval.
\end{enumerate}
\item A person is hypoglycaemic when their blood glucose level is below $0{,}7\,\mathrm{g.L}^{-1}$.\\
\begin{enumerate}
\item[a.] Prove that on the interval $[0;6]$ the equation $f(t) = 0{,}7$ admits a unique solution which we will denote $\alpha$.
\item[b.] How long after having eaten does this person become hypoglycaemic? Express this time to the nearest minute.
\end{enumerate}
\item We wish to determine the average blood glucose level in $\mathrm{g.L}^{-1}$ in this person during the six hours following the meal.\\
\begin{enumerate}
\item[a.] Using integration by parts, show that:
$$\int_0^6 f(t)\,\mathrm{d}t = -23{,}75\,\mathrm{e}^{-2{,}4} + 8{,}75$$
\item[b.] Calculate the average blood glucose level in $\mathrm{g.L}^{-1}$ in this person during the six hours following the meal.
\item[c.] By noting that the function $f$ is a solution of the differential equation (E), explain how we could have obtained this result differently.
\end{enumerate}
\end{enumerate}