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 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 $t , M ( t ) = E ( t ) M \left( t _ { 0 } \right)$.