Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Justify the existence of two solutions $y_1$ and $y_2$ in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) such that: $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \text{ and } \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1. \end{array} \right. \right.$$ Justify that $\operatorname{Vect}(y_1, y_2)$ is the set of solutions of (1) in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.
Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation
$$y'' + qy = 0. \tag{1}$$
Justify the existence of two solutions $y_1$ and $y_2$ in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) such that:
$$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \text{ and } \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1. \end{array} \right. \right.$$
Justify that $\operatorname{Vect}(y_1, y_2)$ is the set of solutions of (1) in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.