grandes-ecoles 2018 Q12

grandes-ecoles · France · centrale-maths2__official Differential equations First-Order Linear DE: General Solution

Let $a, b, r_1$ and $r_2$ be four real numbers such that $0 < r_1 < r_2$. Determine a function $f$ of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$ such that $$\begin{cases} \Delta f = 0 \\ f(x,y) = a & \text{if } \|(x,y)\| = r_1 \\ f(x,y) = b & \text{if } \|(x,y)\| = r_2 \end{cases}$$

Let $a, b, r_1$ and $r_2$ be four real numbers such that $0 < r_1 < r_2$. Determine a function $f$ of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$ such that
$$\begin{cases} \Delta f = 0 \\ f(x,y) = a & \text{if } \|(x,y)\| = r_1 \\ f(x,y) = b & \text{if } \|(x,y)\| = r_2 \end{cases}$$

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