grandes-ecoles 2021 Q26

grandes-ecoles · France · centrale-maths2__psi Differential equations Higher-Order and Special DEs (Proof/Theory)

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ For any real number $x$, express $\Phi_n^{(n)}(x)$ and $\Phi_n^{(n+1)}(x)$ in terms of $L_n(x)$ and $L_n'(x)$.

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$,
$$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$
For any real number $x$, express $\Phi_n^{(n)}(x)$ and $\Phi_n^{(n+1)}(x)$ in terms of $L_n(x)$ and $L_n'(x)$.

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