Ordinary Differential Equations

The aim of the Differential Equations course (EDO) is to teach the basic tools for studying the behavior of solutions of differential equations. After a brief introduction containing examples of differential equations from physics and their use in the context of the engineering profession, we will cover three main chapters: the existence-uniqueness of solutions and the calculation of exact solutions, the stability of solutions to a DDE, and methods for numerically solving DDEs. 

I) Exact solutions of differential equations
1. Definitions: Ordinary differential equations, Solutions of an ODE, Cauchy problems.
2. Existence-Uniqueness of solutions. Grönwall's lemma. Cauchy-Lipschitz theorem.
3. Exact solutions for
- scalar EDOs of order 1 with separable variables,
- linear systems of homogeneous EDOs with constant coefficients
- linear scalar EDOs of order m, homogeneous with constant coefficients.
4. Principles of comparison for the asymptotic study of solutions

II) Asymptotic analysis and stability of solutions
1. Quantities conserved or dissipated and Hamiltonian structure of the equations of mechanics.
2. Lyapunov stability, asymptotic stability, Lyapunov stability theorems.
3. Study the stability of the null solution of a linear ODE system.
4. Study the stability of a stationary solution of a nonlinear ODE by studying the linearized sound spectrum.

III) Numerical methods
1. ODE discretization. Explicit and implicit Euler methods. Other standard methods.
2. Numerical stability and the study of error propagation in the linear case (A-stability).
3. Consistency, Order and Convergence. Numerical convergence theorem.
4. Explicit Runge-Kutta methods
- Presentation of methods and order calculation.
- Stability of a Runge-Kutta method.
5. Multistep linear methods (Adams-Bashforth,...)

1. Analysis of functions of several real variables, regularity of functions and differential calculus.
2. Linear and bilinear algebra. Matrix calculus.
3. Asymptotic analysis, limit calculations and limited developments.
4. Topology of normed vector spaces in finite or infinite dimension.
5. Fundamental theorems of analysis and algebra.