For $0 < \mu \leqslant 1$, we consider $F_\mu$ defined by: $$\forall y \in ]0, +\infty[, \quad F_\mu(y) = \frac{a}{\mu} y\left(1 - \left(\frac{y}{\theta}\right)^\mu\right)$$ and $F_0$ defined by $F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$, with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$.
(a) Show that $F_\mu$ converges pointwise to $F_0$ as $\mu$ tends to 0.
(b) Show that $\phi_\mu$ converges pointwise to $\phi_0$ as $\mu$ tends to 0.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. For all $r \geqslant 0$, we denote $B_r$ the closed ball with center $y_{\text{init}}$ and radius $r$.
Show that we can choose $r > 0$ and $T > 0$ such that $B_r \subset \Omega$ and such that for all $N \in \mathbb{N}^*$, we can define by recursion, by setting $\Delta t = \frac{T}{N}$, a sequence $(y_n)_{0 \leqslant n \leqslant N}$ taking values in $B_r$ such that: $$y_0 = y_{\text{init}}, \quad y_{n+1} = y_n + \Delta t F(y_n), \forall n \in \{0, \cdots, N-1\}$$

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With $r > 0$, $T > 0$, $\Delta t = \frac{T}{N}$ and the sequence $(y_n)_{0 \leqslant n \leqslant N}$ as defined in question III.1, show that we can construct a unique function $\phi_N$ continuous on $[0, T]$, affine on each interval $[n\Delta t, (n+1)\Delta t]$ for all $n \in \{0, \cdots, N-1\}$ and such that $\phi_N(n\Delta t) = y_n$ for all $n \in \{0, \cdots, N\}$.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With the functions $\phi_N$ constructed in question III.2, show, using Theorem 1, that there exists a subsequence of $\phi_N$ that converges uniformly on $[0, T]$ to a continuous function $\phi$.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that we can define a sequence of step functions $\psi_N : [0, T] \rightarrow \mathbb{R}^d$ such that $\psi_N(t) = \phi_N(t)$ for $t \in \{n\Delta t \mid n \in \{0, \cdots, N\}\}$ and such that: $$\forall N \in \mathbb{N}^*, \phi_N(t) = y_{\text{init}} + \int_0^t F(\psi_N(s))\, ds \text{ for all } t \in [0, T]$$ We will specify the expression of the functions $\psi_N$.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With the step functions $\psi_N$ defined in question III.4, deduce that there exists a subsequence of $\psi_N$ that converges uniformly on $[0, T]$ and specify its limit.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that $(\phi, T)$ is a solution of the Cauchy problem $$\left\{\begin{array}{l} y'(t) = F(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$ and deduce the following theorem:
Theorem 2: If $F$ is a continuous function, then there exists at least one solution of the Cauchy problem (1).

We consider the particular case for $d = 1$ given for all $y \in \mathbb{R}$ by $F(y) = 3|y|^{2/3}$ and $y_{\text{init}} = 0$. Show that this Cauchy problem admits infinitely many global solutions.

Mathematics Specialty - EXERCISE II (20 points)

The questions in Part I can be treated independently. In this exercise, $K$ and $a$ are strictly positive real constants.

Part I - Preliminary Studies

Consider the differential equation $\left( E _ { 1 } \right) : z ^ { \prime } ( t ) + z ( t ) = \frac { 1 } { K }$, where $z$ is a function defined and differentiable on $[ 0 ; + \infty [$. II-1- Give the general solution of $( E _ { 1 } )$ on the interval $[ 0 ; + \infty [$. Consider the function $f$ defined for every positive real $t$ by: $f ( t ) = \frac { 10 } { 1 + a e ^ { - t } }$. II-2- Complete the table of variations of $f$ on the interval $[ 0 ; + \infty [$. Specify the value of $f$ at $0$ and the limit of $f$ at $+ \infty$. II-3- Determine, as a function of $a$, the set of solutions of the equation $f ( t ) = 5$.

Part II - Evolution of a Marmot Population

Let $y _ { 0 }$ be a strictly positive real number. We study the evolution of a marmot population, which initially numbers $y _ { 0 }$ thousand individuals. We admit that the size of the population, expressed in thousands of individuals, after $t$ years (with $t \geq 0$) is a function $y$ differentiable on $[ 0 ; + \infty [$, solution of the differential equation: $$\left( E _ { 2 } \right) : y ^ { \prime } ( t ) = y ( t ) \left( 1 - \frac { y ( t ) } { K } \right)$$ The constant $K$ is called the carrying capacity of the environment, expressed in thousands of individuals. We admit that there exists a unique function $y$ solution of $\left( E _ { 2 } \right)$ that satisfies $y ( 0 ) = y _ { 0 }$. We admit that this function takes strictly positive values on $[ 0 ; + \infty [$. We set $z ( t ) = \frac { 1 } { y ( t ) }$ for every positive real $t$. II-4-a- Express $z ^ { \prime } ( t )$ as a function of $y ^ { \prime } ( t )$ and $y ( t )$. II-4-b- We wish to show that $z$ is a solution of $\left( E _ { 1 } \right)$ if, and only if, $y$ is a solution of $\left( E _ { 2 } \right)$. Complete:

• Line 1 using an expression involving $z ^ { \prime } ( t )$ and $z ( t )$;
• Line 2 and Line 3 using an expression involving $y ^ { \prime } ( t )$ and $y ( t )$.

II-5-a- Deduce the general solution of $( E _ { 2 } )$. II-5-b- We admit that the unique solution $y$ of $\left( E _ { 2 } \right)$ satisfying $y ( 0 ) = y _ { 0 }$ is written in the form $y ( t ) = \frac { K } { 1 + a e ^ { - t } }$. Express $a$ as a function of $y _ { 0 }$ and $K$. In a certain valley with carrying capacity $K = 10$, the marmots have disappeared. Scientists wish to reintroduce $y _ { 0 }$ thousand marmots, with $0 < y _ { 0 } < 10$. In the remainder of the exercise, we will take $K = 10$. II-6- Justify that the value of $a$ obtained in question II-5-b- is indeed strictly positive. II-7-a- Using the result from question II-3-, give the value of $a$ such that $y ( 5 ) = 5$. II-7-b- Deduce the exact value of $y _ { 0 }$ such that $y ( 5 ) = 5$. Justify your answer. II-7-c- The calculator gives $0.0669285092$ as the result of the calculation of the value of $y _ { 0 }$ from the previous question. What is the minimum number of marmots to reintroduce so that at least $5$ thousand marmots are present after $5$ years following their reintroduction?

Let $r$ be the function defined by
$$\begin{aligned} r : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show that the function $r$ is of class $\mathcal { C } ^ { 1 }$ on $] - \pi ; \pi [$ and that:
$$\forall \theta \in ] - \pi ; \pi \left[ , \quad r ^ { \prime } ( \theta ) = - \mathrm { i } e ^ { \mathrm { i } \theta } \cdot \int _ { 0 } ^ { + \infty } \frac { t ^ { x } } { \left( 1 + t \mathrm { e } ^ { \mathrm { i } \theta } \right) ^ { 2 } } \mathrm {~d} t . \right.$$
Hint: let $\beta \in ] 0 ; \pi [$, show that for all $\theta \in [ - \beta ; \beta ]$ and $t \in [ 0 , + \infty [$, $\left| 1 + t e ^ { i \theta } \right| ^ { 2 } \geq \left| 1 + t e ^ { i \beta } \right| ^ { 2 } = ( t + \cos ( \beta ) ) ^ { 2 } + ( \sin ( \beta ) ) ^ { 2 }$.

Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show that the function $g$ is of class $\mathcal { C } ^ { 1 }$ on $] - \pi ; \pi [$ and that for all $\theta \in ] - \pi ; \pi [$,
$$g ^ { \prime } ( \theta ) = \mathrm { i } e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } h ^ { \prime } ( t ) \mathrm { d } t$$
where $h$ is the function defined by
$$\begin{aligned} h : ] 0 ; + \infty [ & \longrightarrow \mathbf { C } \\ t & \longmapsto \frac { t ^ { x } } { 1 + t e ^ { \mathrm { i } \theta } } . \end{aligned}$$
Calculate $h ( 0 )$ and
$$\lim _ { t \rightarrow + \infty } h ( t ) .$$
Deduce that the function $g$ is constant on $] - \pi ; \pi [$.

Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$. We define the Laplace transform of $f$ by the function $$\mathcal { L } ( f ) : t \in ] 0 , + \infty \left[ \mapsto \int _ { 0 } ^ { + \infty } e ^ { - t x } f ( x ) d x \right.$$ Prove that $\mathcal { L } ( f )$ is well-defined and of class $\mathcal { C } ^ { 1 }$ on $] 0 , + \infty [$, and express its derivative.

Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$. The purpose of this question is to prove that $$\left( \int _ { 0 } ^ { + \infty } f ( x ) d x \text { converges } \right) \Rightarrow \left( \lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = \int _ { 0 } ^ { + \infty } f ( x ) d x \right)$$
We assume that $\int _ { 0 } ^ { + \infty } f ( x ) d x$ converges.
(a) Prove that the function $$F : x \in \left[ 0 , + \infty \left[ \mapsto \int _ { x } ^ { + \infty } f ( t ) d t \right. \right.$$ is well-defined, continuous and bounded on $\left[ 0 , + \infty \right[$, of class $\mathcal { C } ^ { 1 }$ on $] 0 , + \infty [$ and satisfies $F ^ { \prime } = - f$.
(b) Prove that $\lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = \int _ { 0 } ^ { + \infty } f ( x ) d x$ by integration by parts.

Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$ and $S \in \mathbb { R }$. Prove that $$\left( \lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = S \text { and } f ( t ) \underset { t \rightarrow + \infty } { = } O \left( \frac { 1 } { t } \right) \right) \Rightarrow \left( \int _ { 0 } ^ { + \infty } f ( x ) d x \text { converges and } \int _ { 0 } ^ { + \infty } f ( x ) d x = S \right)$$
For this, using the notations of question 5 of section 2, one can prove that there exist $M > 0$ and $A > 0$ such that for all $t > 0$ $$\begin{aligned} \left| \int _ { A } ^ { + \infty } f ( x ) g \left( e ^ { - t x } \right) d x - \int _ { A } ^ { + \infty } f ( x ) P _ { 1 } \left( e ^ { - t x } \right) d x \right| & \leqslant M \int _ { A } ^ { + \infty } Q \left( e ^ { - t x } \right) e ^ { - t x } \frac { 1 - e ^ { - t x } } { x } d x \\ & \leqslant M \int _ { 0 } ^ { 1 } Q ( u ) d u \end{aligned}$$

Show that if 0 is not a pole of $P/Q \in \mathbf{Q}(x)$, then there exists a unique power series with rational coefficients $g \in \mathbf{Q}\llbracket x \rrbracket$ such that $P = Q \cdot g$.
Show that the map $P/Q \longmapsto g$ is compatible with addition and multiplication in $\mathbf{Q}(x)$ and in $\mathbf{Q}\llbracket x \rrbracket$, and that it sends the derivative $(P/Q)' = (P'Q - PQ')/Q^2$ to the derived power series $g'$.

(More difficult question) Let $f$ be the power series expansion of a rational function $P/Q \in \mathbf{Q}(x)$ all of whose poles are rational numbers. Suppose that the antiderivative $\int_0^x f(t)\,dt$ is globally bounded. Show that $\int_0^x f(t)\,dt$ is then the power series expansion of a rational function in $\mathbf{Q}(x)$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f(x)$ be a function $E$ such that $f(1) = 0$. Show that the power series $f(x)/(x-1)$ is still a function $E$.

Justify that there exists a unique solution $u$ to the Cauchy problem $\left( C _ { \ell } \right)$, give its expression and draw its variation table.
$$\left( C _ { \ell } \right) : \left\{ \begin{array} { l } u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) ) \\ u ( 0 ) = 0 \end{array} . \right.$$

Show that there exists a unique constant solution of equation $\left( E _ { \ell } \right)$, denoted $\gamma \in \mathbf { R }$, and verify that the solution $u$ found in question 1 satisfies
$$\lim _ { x \rightarrow + \infty } u ( x ) = \gamma .$$
where $\left( E _ { \ell } \right) : \quad u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) )$.

Show that $c$ is a constant solution of $(E)$, then that $(E)$ admits exactly two constant solutions denoted $c _ { 1 }$ and $c _ { 2 }$ such that $c _ { 1 } < 0 < c _ { 2 }$. Deduce the value of $c$ as a function of $c _ { 1 }$ and $c _ { 2 }$.
We admit that $y$ is decreasing on $\mathbf { R } _ { + }$ and $\lim _ { x \rightarrow + \infty } y ( x ) = c$, where $c \in \mathbf { R }$. The equation $(E)$ is: $$( E ) : \quad y ^ { \prime } ( x ) + y ( x ) + 1 = \frac { 1 } { 2 } \mathrm { e } ^ { y ( x ) }.$$

Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Show that on $] 0 , \infty [$ we have $f = g$. For this you may use a differential equation satisfied by $( f - g )$ and use the behavior of $f$ and $g$ at $+ \infty$.

Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Using the expression of $g$ obtained in question 2.b., show that $g$ is continuous at 0.

Suppose in this question that $\lambda_1 < \lambda_2 < \cdots < \lambda_n$, and that $J = \{1, 2, \ldots, n\}$. For $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$ we set $$f(x) = \sum_{k=1}^n \frac{\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle^2}{x - \lambda_k}$$ (a) Show that $f$ is of class $C^\infty$ on $\mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, and calculate its derivative $f'(x)$.
(b) Show that the equation $f(x) = 1$ has a unique solution in each interval $]\lambda_\ell, \lambda_{\ell+1}[$ for all $\ell \in \{1, 2, \ldots, n-1\}$, and in $]\lambda_n, +\infty[$.
(c) We denote by $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ the eigenvalues of $B$. Show that $$\lambda_1 < \mu_1 < \lambda_2 < \mu_2 < \cdots < \lambda_n < \mu_n$$

The function $y = f(x)$ is the solution of the differential equation
$$\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^4 + 2x}{\sqrt{1 - x^2}}$$
in $(-1,1)$ satisfying $f(0) = 0$. Then
$$\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x)\, dx$$
is
(A) $\frac{\pi}{3} - \frac{\sqrt{3}}{2}$
(B) $\frac{\pi}{3} - \frac{\sqrt{3}}{4}$
(C) $\frac{\pi}{6} - \frac{\sqrt{3}}{4}$
(D) $\frac{\pi}{6} - \frac{\sqrt{3}}{2}$

Let $y ( x )$ be a solution of the differential equation $\left( 1 + e ^ { x } \right) y ^ { \prime } + y e ^ { x } = 1$. If $y ( 0 ) = 2$, then which of the following statements is (are) true?
(A) $\quad y ( - 4 ) = 0$
(B) $\quad y ( - 2 ) = 0$
(C) $\quad y ( x )$ has a critical point in the interval $( - 1,0 )$
(D) $y ( x )$ has no critical point in the interval $( - 1,0 )$