grandes-ecoles 2020 Q31

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

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. We consider the function $$h : \begin{array}{ccc} ]-R,R[ & \rightarrow & \mathbb{R} \\ x & \mapsto & S(x)\mathrm{e}^{S(x)} \end{array}$$ Prove that $h$ is a solution on $]-R, R[$ of the differential equation $xy' - y = 0$.

We define a sequence $(a_n)_{n \geqslant 1}$ by setting
$$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$
We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. We consider the function
$$h : \begin{array}{ccc} ]-R,R[ & \rightarrow & \mathbb{R} \\ x & \mapsto & S(x)\mathrm{e}^{S(x)} \end{array}$$
Prove that $h$ is a solution on $]-R, R[$ of the differential equation $xy' - y = 0$.

Paper Questions

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