Let $A$ be a continuous function, periodic of period $T > 0$ and $X$ a function of class $\mathcal { C } ^ { 1 }$ $$A : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathcal { M } _ { 2 } ( \mathbb { C } ) \\ & t \mapsto A ( t ) \end{aligned} \quad X \right. : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto \binom { x _ { 1 } ( t ) } { x _ { 2 } ( t ) } \end{aligned}$$ We are interested in the homogeneous differential system with unknown $X$ $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$. We denote by $U$ and $V$ the two solutions of the differential system (IV.1) satisfying $U \left( t _ { 0 } \right) = \binom { 1 } { 0 }$ and $V \left( t _ { 0 } \right) = \binom { 0 } { 1 }$.
We consider the linear differential system (IV.2) whose solutions are functions of class $\mathcal { C } ^ { 1 }$ with values in $\mathcal { M } _ { 2 } ( \mathbb { C } )$ $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ For all $t \in \mathbb { R }$, we set $E ( t ) = [ U ( t ) , V ( t ) ]$. Verify that $E$ is the solution of (IV.2) satisfying $E \left( t _ { 0 } \right) = I _ { 2 }$.

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 $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. Deduce from the previous question that there exists a unique matrix $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ independent of $t$ such that for all real $t , E ( t + T ) = E ( t ) B$.

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 $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the unique matrix such that $E(t+T) = E(t)B$ for all $t$. The Floquet multipliers of (IV.1) are the eigenvalues of $B$.
Let $\rho \in \mathbb { C }$ be a Floquet multiplier of (IV.1) and $Z \in \mathbb { C } ^ { 2 }$ be an eigenvector of $B$ associated with this eigenvalue. We denote $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) Z \end{aligned}$.
a) Prove that $\forall t \in \mathbb { R } , Y ( t + T ) = \rho Y ( t )$.
b) Prove that there exists a complex number $\mu$ and a function $S : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto S ( t ) \end{aligned} \right.$ non-zero and $T$-periodic such that $\forall t \in \mathbb { R } , Y ( t ) = \mathrm { e } ^ { \mu t } S ( t )$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit a non-zero periodic solution of period $T$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Assume that the matrix $B$ is diagonalizable. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit an unbounded solution on $\mathbb { R }$.

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 $W ( t ) = \operatorname { det } ( E ( t ) )$ and denote $\rho _ { 1 }$ and $\rho _ { 2 }$ the Floquet multipliers of (IV.1). Deduce that $\rho _ { 1 } \rho _ { 2 } = \exp \left( \int _ { 0 } ^ { T } \operatorname { tr } ( A ( s ) ) \mathrm { d } s \right)$.

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 if $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a solution of (1), then the function $t \mapsto y(t+T)$ is also one. Deduce that for all $t \in \mathbb{R}$: $$y(t+T) = y(T) y_1(t) + y'(T) y_2(t)$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Let $y_1$ and $y_2$ be the solutions in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) satisfying $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Let $\mu \in \mathbb{C}^*$, and let $\lambda \in \mathbb{C}$ such that $\mu = e^{\lambda T}$. Show that the following three assertions are equivalent.
(a) Equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ that satisfies: $$\forall t \in \mathbb{R}, \quad y(t+T) = \mu y(t)$$
(b) The complex number $\mu$ is a solution of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0$$
(c) The differential equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad y(t) = e^{\lambda t} u(t)$$ where $u \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a $T$-periodic function.

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 $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0$$ Show that if $\mu_1 \neq \mu_2$ and if $\lambda$ is a complex number such that $\mu_1 = e^{\lambda T}$, then for any solution $y$ of (1), there exist two $T$-periodic functions $w_1$ and $w_2$, as well as two complex numbers $\alpha$ and $\beta$ such that $$\forall t \in \mathbb{R}, \quad y(t) = \alpha e^{\lambda t} w_1(t) + \beta e^{-\lambda t} w_2(t)$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Let $y_1$ and $y_2$ be the solutions in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) satisfying $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0$$ Suppose that $\mu_1 = \mu_2$. Show that $\mu_1 = \mu_2 = \pm 1$ and that equation (1) admits a periodic solution in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

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 $\mathbb{R}$.
Show that there exists $\mu \in \mathbb{C}^*$ and a non-zero solution $Y \in \mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2) such that $$\forall t \in \mathbb{R}, \quad Y(t+T) = \mu Y(t)$$

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 the matrix $(M(t))^{-1} M(t+T)$ is independent of $t \in \mathbb{R}$.

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 $Y_1(t), \ldots, Y_n(t)$.
Deduce that there exists $B \in \mathscr{M}_n(\mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad M(t+T) = M(t) \exp(TB)$$

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 $Y_1(t), \ldots, Y_n(t)$. There exists $B \in \mathscr{M}_n(\mathbb{C})$ such that $M(t+T) = M(t)\exp(TB)$ for all $t \in \mathbb{R}$.
Deduce that there exists an application $Q : \mathbb{R} \rightarrow \mathrm{GL}_n(\mathbb{C})$ continuous on $\mathbb{R}$ and $T$-periodic such that $$\forall t \in \mathbb{R}, \quad M(t) = Q(t) \exp(tB)$$ (This identity is called the normal form of the matrix $M$).

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 $$X'(t) = A(t) X(t) \tag{2}$$ 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 $t \in \mathbb{R}$, let $M(t)$ be the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$, and let $M(t) = Q(t)\exp(tB)$ be the normal form. We admit that there exist two matrices $D$ and $N$ of $\mathscr{M}_n(\mathbb{C})$ such that $D$ is diagonalizable, $N$ is nilpotent and $B = D + N$ and $DN = ND$. There exists a matrix $P \in \mathrm{GL}_n(\mathbb{C})$ and a diagonal matrix $\Delta$ such that $D = P\Delta P^{-1}$.
For $t \in \mathbb{R}$, we denote by $Z_1(t), Z_2(t), \ldots, Z_n(t) \in \mathbb{C}^n$ the columns of the matrix $M(t)P$. Show that $(Z_1, Z_2, \ldots, Z_n)$ is a basis of the space $\mathscr{S}$.

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 $$X'(t) = A(t) X(t) \tag{2}$$ 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 $t \in \mathbb{R}$, let $M(t)$ be the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$, and let $M(t) = Q(t)\exp(tB)$ be the normal form. We admit that $B = D + N$ where $D$ is diagonalizable, $N$ is nilpotent, and $DN = ND$. There exists $P \in \mathrm{GL}_n(\mathbb{C})$ and a diagonal matrix $\Delta$ such that $D = P\Delta P^{-1}$. Let $Z_1(t), Z_2(t), \ldots, Z_n(t)$ be the columns of $M(t)P$.
Let $\lambda_1, \ldots, \lambda_n$ be the complex numbers such that $\Delta = \operatorname{Diag}(\lambda_1, \lambda_2, \ldots, \lambda_n)$. For all $0 \leqslant i \leqslant n-1$, $1 \leqslant k \leqslant n$ and $t \in \mathbb{R}$, we denote by $R_{i,k}(t)$ the $k$-th column of the matrix $\frac{1}{i!} Q(t) N^i P$. Show that for all $k \in \{1, 2, \ldots, n\}$, we have $$Z_k(t) = e^{\lambda_k t} \left(\sum_{i=0}^{n-1} t^i R_{i,k}(t)\right)$$ and verify that the applications $R_{i,k}$ are continuous on $\mathbb{R}$ and $T$-periodic.

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 $$X'(t) = A(t) X(t) \tag{2}$$ Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$.
Show that if $B$ has an eigenvalue of the form $\lambda = i\frac{2k\pi}{mT}$ with $k \in \mathbb{Z}$ and $m \in \mathbb{N}^*$, then (2) has a non-zero $mT$-periodic solution.

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 $$X'(t) = A(t) X(t) \tag{2}$$ Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$.
Suppose that there exists $m \in \mathbb{N}^*$ such that (2) has a non-zero $mT$-periodic solution. Show that $\exp(TB)$ has an eigenvalue that is an $m$-th root of unity.

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 $$X'(t) = A(t) X(t) \tag{2}$$ In this question, we suppose that (2) has a $T'$-periodic solution $X$ with $T' \notin \mathbb{Q} T$.
Show that for all $t \in \mathbb{R}$ and $u \in \mathbb{R}$, we have $$A(u) X(t) = A(t) X(t).$$ One may use without proof the fact that if $G$ is a subgroup of $(\mathbb{R}, +)$ which is not of the form $\mathbb{Z}a$ for $a \in \mathbb{R}$, then $G$ is dense in $\mathbb{R}$.

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 $$X'(t) = A(t) X(t) \tag{2}$$ Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$.
We assume that there does not exist a vector subspace $V \subset \mathbb{C}^n$, different from $\{0\}$ and $\mathbb{C}^n$, such that, for all $t \in \mathbb{R}$, $V$ is stable under $A(t)$. Give a necessary and sufficient condition on $A$ and on $B$ for (2) to have at least one non-zero periodic solution.

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. Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$. Let the differential system $$X'(t) = A(t) X(t) + b(t) \tag{3}$$ where $b : \mathbb{R} \rightarrow \mathbb{C}^n$ is a continuous function on $\mathbb{R}$ and $T$-periodic.
We assume that $1$ is not an eigenvalue of $\exp(TB)$. Show that (3) possesses a unique $T$-periodic solution.

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 if $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a solution of (1), then the function $t \mapsto y(t+T)$ is also one. Deduce that for all $t \in \mathbb{R}$: $$y(t+T) = y(T) y_1(t) + y'(T) y_2(t).$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Let $\mu \in \mathbb{C}^*$, and let $\lambda \in \mathbb{C}$ such that $\mu = e^{\lambda T}$. Show that the following three assertions are equivalent.
(a) Equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ that satisfies: $$\forall t \in \mathbb{R}, \quad y(t+T) = \mu y(t).$$
(b) The complex number $\mu$ is a solution of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$
(c) The differential equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad y(t) = e^{\lambda t} u(t),$$ where $u \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a $T$-periodic function.

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 $y_2(0)=0, y_2'(0)=1$. Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$ Show that if $\mu_1 \neq \mu_2$ and if $\lambda$ is a complex number such that $\mu_1 = e^{\lambda T}$, then for any solution $y$ of (1), there exist two $T$-periodic functions $w_1$ and $w_2$, as well as two complex numbers $\alpha$ and $\beta$ such that $$\forall t \in \mathbb{R}, \quad y(t) = \alpha e^{\lambda t} w_1(t) + \beta e^{-\lambda t} w_2(t).$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$ Suppose that $\mu_1 = \mu_2$. Show that $\mu_1 = \mu_2 = \pm 1$ and that equation (1) admits a periodic solution in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.