grandes-ecoles 2018 Q16

grandes-ecoles · France · centrale-maths2__mp Differential equations First-Order Linear DE: General Solution

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. Solve (II.1) on $\mathbb{R}^{+*}$: $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set
$$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$
We assume here that $\lambda = 0$. Solve (II.1) on $\mathbb{R}^{+*}$:
$$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$

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