grandes-ecoles 2021 Q29

grandes-ecoles · France · centrale-maths2__psi Differential equations Verification that a Function Satisfies a DE

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Deduce that $L_n$ is a solution of the differential equation $$x L_n''(x) + (1-x) L_n'(x) + n L_n(x) = 0.$$

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$,
$$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$
Deduce that $L_n$ is a solution of the differential equation
$$x L_n''(x) + (1-x) L_n'(x) + n L_n(x) = 0.$$

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