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 $T > 0$ 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 $E \left( t _ { 0 } \right) = I _ { 2 }$.