grandes-ecoles 2020 Q15

grandes-ecoles · France · centrale-maths2__mp Second order differential equations Second-order ODE with initial or boundary value conditions

For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$ Let $\lambda \in \mathbb{R}$ be a nonzero eigenvalue of $T$ and $f$ be an associated eigenvector. Show that $f$ is a solution of the differential equation $\lambda f'' = -f$.

For all $f \in E$, we set,
$$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$
Let $\lambda \in \mathbb{R}$ be a nonzero eigenvalue of $T$ and $f$ be an associated eigenvector. Show that $f$ is a solution of the differential equation $\lambda f'' = -f$.

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