We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
Using question II.D, show by induction that for all integer $n \geqslant 1$ the function $\varphi _ { n }$ is strictly positive on $] 0 , \alpha _ { 0 } [$.

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$.
Justify that there exists a real $A > 0$ such that for $x > A , q ( x ) > c ^ { 2 }$ ($q$ defined in III.A.2).

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$.
Let $a > A$. We set for $x > 0 , z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, solution of (IV.1). We define the function $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$.
Verify that for $x > 0 , W ^ { \prime } ( x ) = \left( q ( x ) - c ^ { 2 } \right) z ( x ) z _ { 1 } ( x )$.

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$. Let $a > A$, $z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, and $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$.
We denote $\left. I _ { a } = \right] a , a + \pi / c \left[ \right.$ and assume that $\varphi _ { n }$ has no zeros on $I _ { a }$.
Determine the signs of $W ( a ) , W ( a + \pi / c )$ and of $W ^ { \prime }$ on $I _ { a }$ and reach a contradiction. Deduce that $\varphi _ { n }$ has a zero in every interval $I _ { a }$ with $a > A$.
One may distinguish cases according to the sign of $\varphi _ { n }$ on $I _ { a }$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ In other words, $u_a$ is the unique function of class $C^1$ from $\mathbb{R}$ to $\mathbb{R}^2$ such that $u_a(0) = a$ and, for every real $t$, $u_a'(t) = A u_a(t)$.
We assume $A$ is diagonal of the form $$A = \operatorname{diag}(\lambda_1, \lambda_2) = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$
What is $u_a(t)$?

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ In other words, $u_a$ is the unique function of class $C^1$ from $\mathbb{R}$ to $\mathbb{R}^2$ such that $u_a(0) = a$ and, for every real $t$, $u_a'(t) = A u_a(t)$.
We assume $A$ is diagonal of the form $$A = \operatorname{diag}(\lambda_1, \lambda_2) = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$
Let $a$ and $b$ be two elements of $\mathbb{R}^2$ and let $t$ be a real number. Show that $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ In other words, $u_a$ is the unique function of class $C^1$ from $\mathbb{R}$ to $\mathbb{R}^2$ such that $u_a(0) = a$ and, for every real $t$, $u_a'(t) = A u_a(t)$.
We assume $A$ is diagonal of the form $$A = \operatorname{diag}(\lambda_1, \lambda_2) = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$
Use the result of II.B.2 to interpret the sign of $\operatorname{div}_f(a)$ in terms of the direction of variation of the area of a certain parallelogram as a function of $t$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ We still assume that $A = \operatorname{diag}(\lambda_1, \lambda_2)$.
We set $a = (a_1, a_2)$ and $u_a(t) = (x_1(t), x_2(t))$. We assume that $\lambda_1 \neq 0$ and $a_1 > 0$. Determine a function $\theta_a$ such that $x_2(t) = \theta_a(x_1(t))$ for every real $t$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ We still assume that $A = \operatorname{diag}(\lambda_1, \lambda_2)$.
In this question, $a = (2,1)$ and $b = (1,2)$.
For each of the following cases, illustrate on the same figure the graphs of the functions $\theta_a$, $\theta_b$ and $\theta_{a+b}$, as well as the parallelograms with vertices $(0,0)$, $u_a(t)$, $u_b(t)$ and $u_a(t) + u_b(t)$ for $t = 0$ and a strictly positive value of $t$.
a) $\lambda_1 = 1$ and $\lambda_2 = 2$.
b) $\lambda_1 = 1$ and $\lambda_2 = -2$.
c) $\lambda_1 = 1$ and $\lambda_2 = -1$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$
Revisit questions II.B.1 and II.B.2 in the case where $A$ is triangular of the form $$A = \begin{pmatrix} \lambda & \mu \\ 0 & \lambda \end{pmatrix}$$

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$
Show that the relation $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$ holds when the matrix $A$ has a characteristic polynomial that splits over $\mathbb{R}$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$
Extend the result $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$ to the case of an arbitrary real $2 \times 2$ matrix.

We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Show that if $\lambda \in \sigma_p(T)$ and $f \in \operatorname{Ker}(T - \lambda Id)$, then $f \in C^2([0,1], \mathbb{R})$ and satisfies the equation $$\lambda f'' + f = 0$$ with the conditions $f(0) = f(1) = 0$.

We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Deduce $\sigma_p(T)$. Calculate the eigenspaces $E_{\lambda} = \operatorname{Ker}(T - \lambda Id)$ associated with each element $\lambda \in \sigma_p(T)$.

Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
We set $F_{1} = \Re(f)$ and $f_{2} = \Im(f)$. Express $f_{1}^{\prime\prime}$ and $f_{2}^{\prime\prime}$ in terms of $f_{1}^{\prime}, f_{2}^{\prime}, f_{1}$ and $f_{2}$.

Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
Show that $$\forall x \in \mathbb{R}, \left|f^{\prime}(x)\right|^{2} + \frac{1}{4\alpha}\left(\alpha|f(x)|^{2} + 1\right)^{2} = \frac{1}{4\alpha}(\alpha + 1)^{2}$$

Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
Show that $\forall x \in \mathbb{R}, |f(x)| \leq 1$ and that $$\forall x \in \mathbb{R}, \left|f^{\prime}(x)\right| \leq \frac{1}{2\sqrt{\alpha}}(\alpha + 1)$$

Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
The purpose of this question is to prove that there exist $(\ell, M_{0}) \in \mathbb{R}_{+}^{2}$ such that $$\forall x > 0, \left||f(x)|^{2} - 1\right| \leq \frac{M_{0}}{x}$$
(a) Show that $$\forall x \in \mathbb{R}, \Im\left(f^{\prime}(x)\overline{f(x)}\right) + \frac{x}{4}|f(x)|^{2} - \frac{1}{4}\int_{0}^{x}|f(t)|^{2}\,dt = 0$$
(b) Show that $$\forall x > 0, \frac{d}{dx}\left(\frac{1}{x}\int_{0}^{x}|f(t)|^{2}\,dt\right) = -\frac{4}{x^{2}}\Im\left(f^{\prime}(x)\overline{f(x)}\right)$$
(c) Deduce that there exists $\ell \in \mathbb{R}_{+}$ such that $$\lim_{x \rightarrow +\infty} \frac{1}{x}\int_{0}^{x}|f(t)|^{2}\,dt = \ell$$
(d) Show that there exists $M \in \mathbb{R}_{+}$ such that $$\forall x > 0, \left|\frac{1}{x}\int_{0}^{x}|f(t)|^{2}\,dt - \ell\right| \leq \frac{M}{x}$$
(e) Conclude.

Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
(a) Suppose in this question that $\ell = 1$. Show that there exists $M_{1} \in \mathbb{R}_{+}$ such that $$\forall x > 0, \left||f(x)|^{2} - 1\right| \leq \frac{M_{1}}{x^{3/2}}$$
(b) Deduce that $\ell < 1$.

Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
Show that $|f|$ is not periodic.

Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
For $(\ell, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}$, we set: $$f_{\alpha}(x) = f(x)\exp\left(i\frac{x^{2}}{4}\right), \quad \Psi_{\alpha}(t, x) = \frac{1}{\sqrt{t}} f_{\alpha}\left(\frac{x}{\sqrt{t}}\right)$$
(a) Does there exist $t > 0$ such that $\Psi_{\alpha}(t, .)$ is periodic?
(b) Express $f_{\alpha}^{\prime}, f_{\alpha}^{\prime\prime}$ and $|f_{\alpha}|$ in terms of $f, f^{\prime}, f^{\prime\prime}$ and $|f|$.
(c) Justify that for all $(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}$, we have $\Psi_{\alpha}(., x) \in C^{1}(\mathbb{R}^{+*}, \mathbb{C})$ and $\Psi_{\alpha}(t, .) \in C^{2}(\mathbb{R}, \mathbb{C})$, then prove that $\Psi_{\alpha}$ satisfies equation $(F_{\alpha})$: $$\left(F_{\alpha}\right) \quad i\frac{\partial \Psi_{\alpha}}{\partial t}(t, x) + \frac{\partial^{2} \Psi_{\alpha}}{\partial x^{2}}(t, x) + \frac{1}{2}\Psi_{\alpha}(t, x)\left(\alpha\left|\Psi_{\alpha}(t, x)\right|^{2} + \frac{1}{t}\right) = 0$$

Let $\left(a_{k}\right)_{k \in \mathbb{N}}$ be a sequence of complex terms such that the series with general term $k^{2} a_{k}$ is absolutely convergent. For $(t, x) \in \mathbb{R}^{2}$ we then denote $$\Phi_{0}(t, x) = \sum_{k=0}^{+\infty} a_{k} e^{-ik^{2}t + ikx}$$
(a) Show that $\Phi_{0}$ is well defined on $\mathbb{R}^{2}$.
(b) Show that for all $(t, x) \in \mathbb{R}^{2}$ we have $\Phi_{0}(., x) \in C^{1}(\mathbb{R}, \mathbb{C})$ and $\Phi_{0}(t, .) \in C^{2}(\mathbb{R})$. Calculate $\frac{\partial \Phi_{0}}{\partial t}(t, x)$ and $\frac{\partial^{2} \Phi_{0}}{\partial x^{2}}(t, x)$.
(c) Let $(c_{k})$ be a sequence of complex terms such that the series with general term $k^{2} c_{k}$ is absolutely convergent. For $x \in \mathbb{R}$, we set $$f_{0}(x) = \sum_{k=0}^{+\infty} c_{k} e^{ikx}$$ Construct a function $\Psi_{0}$ defined on $\mathbb{R}_{+}^{*} \times \mathbb{R}$ which satisfies

• For all $(t, x) \in \mathbb{R}_{+}^{*}$, $\Psi_{0}(., x) \in C^{1}(\mathbb{R}_{+}^{*}, \mathbb{C})$ and $\Psi_{0}(t, .) \in C^{2}(\mathbb{R}, \mathbb{C})$ and $\Psi_{0}$ is a solution of equation $(F_{0})$, that is $$\forall (t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}, \quad i\frac{\partial \Psi_{0}}{\partial t}(t, x) + \frac{\partial^{2} \Psi_{0}}{\partial x^{2}}(t, x) + \frac{1}{2t}\Psi_{0}(t, x) = 0$$
• For all $t > 0$, $\Psi_{0}(t, .)$ is periodic.
• For all $x \in \mathbb{R}$, $\Psi_{0}(1, x) = f_{0}(x)$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ and $F(t)$ denotes the limit of $F_n(t) = I_3 + \sum_{k=1}^n \frac{t^k \mathcal{M}^k}{k!}$ as defined in question 9.
Show that $F \in C^{1}(\mathbb{R}, \mathcal{M}_{3}(\mathbb{R}))$ and that for all $t \in \mathbb{R}$, we have $F^{\prime}(t) = F(t)\mathcal{M}$.

We assume that $m = 0$ and there exists $\lambda > 0$ such that $G(0) = (0,0,2\lambda)$, $G^{\prime}(0) = (1,0,0)$, and $G$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime\prime}(x) + \left(\lambda^{2} + \frac{x^{2}}{4}\right) G^{\prime}(x) - \frac{x}{4} G(x) = 0$$
(a) Write the linear differential equation $Y^{\prime\prime\prime} + \left(\lambda^{2} + \frac{x^{2}}{4}\right) Y^{\prime} - \frac{x}{4} Y = 0$, where $Y \in C^{3}(\mathbb{R}, \mathbb{R}^{3})$, in the form of a differential system $X^{\prime} = AX$, where $X \in C^{1}(\mathbb{R}, \mathcal{M}_{n,1}(\mathbb{R}))$ and where $A \in C(\mathbb{R}, \mathcal{M}_{n}(\mathbb{R}))$, with $n \in \mathbb{N}^{*}$. We will specify $n$ and $A$.
(b) Show that the coordinates $G_{1}, G_{2}, G_{3}$ of $G$ satisfy $$\forall x \in \mathbb{R}, G_{1}(-x) = -G_{1}(x),\quad G_{2}(-x) = G_{2}(x),\quad G_{3}(-x) = G_{3}(x)$$
(c) Show that for all $x \in \mathbb{R}$, $\|G(x)\|^{2} = x^{2} + 4\lambda^{2}$.
(d) Establish that if $G_{1}$ does not vanish on $\mathbb{R}^{*}$, then $G$ is an injective application on $\mathbb{R}$.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Prove that $F$ satisfies on $\mathbb{R}$ a differential equation of the form $F^{\prime} + AF = 0$, where $A$ is a function to be specified.