grandes-ecoles 2010 QII.D

grandes-ecoles · France · centrale-maths1__psi Taylor series Taylor's formula with integral remainder or asymptotic expansion

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
Let $m$ be the maximal solution determined in question II.B.3).
II.D.1) Show that the solution $m$ is expandable as a power series in a neighborhood of 0. Calculate this expansion and specify its radius of convergence. II.D.2) Deduce the power series expansions of all maximal solutions of $(E)$; specify the radii of convergence of these power series.

We study the differential equation
$$y(x) y'(x) = -4x \tag{E}$$

Let $m$ be the maximal solution determined in question II.B.3).

II.D.1) Show that the solution $m$ is expandable as a power series in a neighborhood of 0. Calculate this expansion and specify its radius of convergence.\\
II.D.2) Deduce the power series expansions of all maximal solutions of $(E)$; specify the radii of convergence of these power series.

Paper Questions

QI.A QI.B QI.C QI.D QI.E QII.A QII.B QII.C QII.D QIII.A QIII.B QIII.C QIII.D