grandes-ecoles 2015 Q11

grandes-ecoles · France · x-ens-maths__psi Second order differential equations Structure of the solution space

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ and $F(t)$ denotes the limit of $F_n(t) = I_3 + \sum_{k=1}^n \frac{t^k \mathcal{M}^k}{k!}$ as defined in question 9.
Show that $F \in C^{1}(\mathbb{R}, \mathcal{M}_{3}(\mathbb{R}))$ and that for all $t \in \mathbb{R}$, we have $F^{\prime}(t) = F(t)\mathcal{M}$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix
$$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
and $F(t)$ denotes the limit of $F_n(t) = I_3 + \sum_{k=1}^n \frac{t^k \mathcal{M}^k}{k!}$ as defined in question 9.

Show that $F \in C^{1}(\mathbb{R}, \mathcal{M}_{3}(\mathbb{R}))$ and that for all $t \in \mathbb{R}$, we have $F^{\prime}(t) = F(t)\mathcal{M}$.

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