### Concours X: mathématiques B MP

Here's a correction of the second Maths composition in MP for the X competitive entrance exam.

It's a very difficult subject, in 4 parts.

The main object is the study of linear differential equations with periodic coefficients: we are looking for class \(\mathscr{C}^1\) functions \(X:\mathbb{R}\to\mathbb{C}^n\) which satisfy \[ \begin{alignat*}{1} X'(t)&= A(t)X(t) \end{alignat*} \]

where \[ \begin{alignat*}{1} A:\mathbb{R}&\to M_n(\mathbb{C})\\ t&\mapsto A(t) \end{alignat*} \] is a continuous function, \(T\) periodic.

In particular, we study conditions that ensure the existence of periodic solutions.

In the last question we solve \(X'=AX\) with \[ A(t) = \begin{bmatrix} 1 & -\cos(t)\\ \cos(t) & 1 \end{bmatrix} \]

Answers to the Maths B test on April 16 in MP concours X ENS 2024:

The problem statement is here: