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 $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We consider the linear differential system $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ If $M : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathcal { M } _ { 2 } ( \mathbb { C } ) \\ & t \mapsto [ F ( t ) , G ( t ) ] \end{aligned} \right.$ is a solution of (IV.2) and $W = \binom { w _ { 1 } } { w _ { 2 } } \in \mathbb { C } ^ { 2 }$, prove that the function $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto M ( t ) W = w _ { 1 } F ( t ) + w _ { 2 } G ( t ) \end{aligned}$ is a solution of (IV.1).

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$. We denote by $U$ and $V$ the two solutions of (IV.1) satisfying $U \left( t _ { 0 } \right) = \binom { 1 } { 0 }$ and $V \left( t _ { 0 } \right) = \binom { 0 } { 1 }$, and set $E ( t ) = [ U ( t ) , V ( t ) ]$.
Let $t _ { 1 } \in \mathbb { R }$ and $W = \binom { w _ { 1 } } { w _ { 2 } } \in \mathbb { C } ^ { 2 }$. Assume that $E \left( t _ { 1 } \right) W = \binom { 0 } { 0 }$. Show that the function $Y : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) W = w _ { 1 } U ( t ) + w _ { 2 } V ( t ) \end{aligned} \right.$ is zero. Deduce that for all real $t , E ( t )$ is invertible.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We consider the linear differential system $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ Let $M \in \mathcal { C } ^ { 1 } \left( \mathbb { R } , \mathcal { M } _ { 2 } ( \mathbb { C } ) \right)$ be a solution of system (IV.2). Show that for all real $t , M ( t ) = E ( t ) M \left( t _ { 0 } \right)$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set for all $t \in \mathbb { R } , W ( t ) = \operatorname { det } ( E ( t ) )$. Show that for all real $t , W ^ { \prime } ( t ) = \operatorname { tr } ( A ( t ) ) W ( t )$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
Let $T \in M_{n}(\mathbf{R})$. We assume that $M$ is similar to $T$ in $M_{n}(\mathbf{R})$ and we denote by $(\mathrm{S}^{*})$ the differential system $$(\mathrm{S}^{*}) \quad Y' = TY$$
Prove that the coordinates of a solution $X$ of (S) are linear combinations of the coordinates of a solution $Y$ of $(\mathrm{S}^{*})$.

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$, inclusions $q_i$, $a_i = p_i a q_i$, and $a = \sum _ { i = 1 } ^ { r } q _ { i } a _ { i } p _ { i }$.
Deduce that: $$\forall t \in \mathbf { R } , \quad e ^ { t a } = \sum _ { i = 1 } ^ { r } q _ { i } e ^ { t a _ { i } } p _ { i }$$

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 $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that for every real number $t$, $M(t) \in \mathrm{GL}_n(\mathbb{C})$ and $M'(t) = A(t) M(t)$.

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: $$\forall t \in \mathbb{R}, \quad y_1(t) y_2'(t) - y_1'(t) y_2(t) = 1.$$

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be an application continuous on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that for every real number $t$, $M(t) \in \mathrm{GL}_n(\mathbb{C})$ and $M'(t) = A(t) M(t)$.