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 $\mu \in \mathbb{C}^*$, 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.