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 $x$:
$$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$
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
$$\forall t \in \mathbb{R}, \quad y(t) = \alpha e^{\lambda t} w_1(t) + \beta e^{-\lambda t} w_2(t).$$