\section*{Exercise 1}
\textbf{PART A}\\
Consider the differential equation
$$( E ) : \quad y ^ { \prime } + \frac { 1 } { 4 } y = 20 \mathrm { e } ^ { - \frac { 1 } { 4 } x } ,$$
with unknown $y$, a function defined and differentiable on the interval $[ 0 ; + \infty [$.

\begin{enumerate}
  \item Determine the value of the real number $a$ such that the function $g$ defined on the interval $[ 0 ; + \infty [$ by $g ( x ) = a x \mathrm { e } ^ { - \frac { 1 } { 4 } x }$ is a particular solution of the differential equation $( E )$.
  \item Consider the differential equation
$$\left( E ^ { \prime } \right) : \quad y ^ { \prime } + \frac { 1 } { 4 } y = 0 ,$$
with unknown $y$, a function defined and differentiable on the interval $[ 0 ; + \infty [$.\\
Determine the solutions of the differential equation ( $E ^ { \prime }$ ).
  \item Deduce the solutions of the differential equation ( $E$ ).
  \item Determine the solution $f$ of the differential equation ( $E$ ) such that $f ( 0 ) = 8$.
\end{enumerate}

\textbf{PART B}\\
Consider the function $f$ defined on the interval $[ 0 ; + \infty [$ by
$$f ( x ) = ( 20 x + 8 ) \mathrm { e } ^ { - \frac { 1 } { 4 } x }$$
It is admitted that the function $f$ is differentiable on the interval $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function on the interval $\left[ 0 ; + \infty \left[ \right. \right.$. Moreover, it is admitted that $\lim _ { x \rightarrow + \infty } f ( x ) = 0$.

\begin{enumerate}
  \item a. Justify that, for every positive real number $x$,
$$f ^ { \prime } ( x ) = ( 18 - 5 x ) \mathrm { e } ^ { - \frac { 1 } { 4 } x }$$
b. Deduce the table of variations of the function $f$. The exact value of the maximum of the function $f$ on the interval $[ 0 ; + \infty [$ will be specified.
  \item In this question we are interested in the equation $f ( x ) = 8$.\\
a. Justify that the equation $f ( x ) = 8$ admits a unique solution, denoted $\alpha$, in the interval [14; 15].\\
b. Copy and complete the table below by running step by step the solution\_equation function opposite, written in Python language

\begin{center}
\begin{tabular}{ | l | c | l | l | l | l | }
\hline
$a$ & 14 &  &  &  &  \\
\hline
$b$ & 15 &  &  &  &  \\
\hline
$b - a$ & 1 &  &  &  &  \\
\hline
$m$ & 14,5 &  &  &  &  \\
\hline
\begin{tabular}{ l }
Condition \\
$f ( m ) > 8$ \\
\end{tabular} & FALSE &  &  &  &  \\
\hline
\end{tabular}
\end{center}

\begin{verbatim}
from math import exp
def f(x) :
    return (20* x +8)*exp(-1/4* x)
def solution_equation() :
    a,b = 14,15
    while b-a>0.1:
        m = (a+b)/2
        if f (m) > 8 :
        |a=m
        else :
        | b=m
    return a,b
\end{verbatim}

c. What is the objective of the solution\_equation function in the context of the question?
\end{enumerate}