grandes-ecoles 2018 Q19

grandes-ecoles · France · centrale-maths2__official Second order differential equations Euler-type (Cauchy-Euler) second-order ODE

We now assume $\lambda \neq 0$. Solve the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ on $\mathbb{R}^{+*}$. One may consider, justifying its existence, a function $Z$ of class $\mathcal{C}^2$ on $\mathbb{R}$ such that, for all $r > 0$, $z(r) = Z(\ln(r))$.

We now assume $\lambda \neq 0$. Solve the differential equation (II.1)
$$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$
on $\mathbb{R}^{+*}$. One may consider, justifying its existence, a function $Z$ of class $\mathcal{C}^2$ on $\mathbb{R}$ such that, for all $r > 0$, $z(r) = Z(\ln(r))$.

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