We consider the differential equation: $$\forall t \in \left[ 0 , + \infty \left[ , A x ^ { \prime \prime } ( t ) = - K x ( t ) \right. \right. \tag{1}$$ with unknown function $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$.
Show that $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ is a solution of the differential equation (1) if and only if there exist $2 n$ real numbers $\left( a _ { i } \right) _ { 1 \leq i \leq n } , \left( b _ { i } \right) _ { 1 \leq i \leq n }$ such that: $$\forall t \in \left[ 0 , + \infty \left[ , x ( t ) = \sum _ { i = 1 } ^ { n } \left( a _ { i } \cos \left( t \sqrt { \lambda _ { i } } \right) + b _ { i } \sin \left( t \sqrt { \lambda _ { i } } \right) \right) e _ { i } \right. \right.$$ Deduce that the set of solutions of (1) is a finite-dimensional vector space and specify its dimension.

Let $g \in C_{b}(\mathbb{R})$. We say that $g$ satisfies hypothesis A if $g$ is a function of class $C^{\infty}$ on $\mathbb{R}$, bounded and whose derivative functions of all orders are bounded on $\mathbb{R}$. Show that if $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$ and if $g$ satisfies hypothesis A, then $g$ is a solution of a linear differential equation with constant coefficients.

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 $F^{\prime}(t) = F(t)\mathcal{M}$.

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Justify the existence of two solutions $y_1$ and $y_2$ in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) such that: $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Justify that $\operatorname{Vect}(y_1, y_2)$ is the set of solutions of (1) in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Let $y_1$ and $y_2$ be the solutions in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) satisfying $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Show that: $$\forall t \in \mathbb{R}, \quad y_1(t) y_2'(t) - y_1'(t) y_2(t) = 1$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Justify the existence of two solutions $y_1$ and $y_2$ in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) such that: $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \text{ and } \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1. \end{array} \right. \right.$$ Justify that $\operatorname{Vect}(y_1, y_2)$ is the set of solutions of (1) in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Show that: $$\forall t \in \mathbb{R}, \quad y_1(t) y_2'(t) - y_1'(t) y_2(t) = 1.$$