grandes-ecoles 2025 Q3

grandes-ecoles · France · x-ens-maths__psi Differential equations Solving Separable DEs with Initial Conditions

In this question only, we set $f(x) := \frac{1}{2}Lx^2$ for all $x \in \mathbb{R}$, where $L > 0$ is fixed. a) Show that $x_{n+1} = (1 - \tau L)x_n$, then express directly $x_n$ as a function of $x_0$ and $n$. b) We suppose $x_0 \neq 0$. Justify that $x_n \rightarrow 0$ if and only if $0 < \tau < 2/L$.

In this question only, we set $f(x) := \frac{1}{2}Lx^2$ for all $x \in \mathbb{R}$, where $L > 0$ is fixed.\\
a) Show that $x_{n+1} = (1 - \tau L)x_n$, then express directly $x_n$ as a function of $x_0$ and $n$.\\
b) We suppose $x_0 \neq 0$. Justify that $x_n \rightarrow 0$ if and only if $0 < \tau < 2/L$.

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