The course introduces some of the most important results in the theory of ordinary differential equations : existence et uniqueness of the Cauchy problem, linear equations, stability of equilibria and resolution methods for some nonlinear equations.
This course introduces the concept of differential equation. It presents some interesting theoretical aspects for a physicist (notion of Cauchy problem, existence and uniqueness of a solution, stability of a fixed point, numerical integration techniques) but especially the different traditional techniques of solving ordinary differential equations (linear ODE of the 1st order, method of variation of constants, ODE with separable variables, homogeneous, Bernouilli, linear ODE of the 2nd order, systems of linear ODE, resonance phenomena).
Chapter I. Introduction and first definitions.
Chapter II. Cauchy's problem.
(Chapter III. Extension of solutions).
(Chapter IV. Continued Dependence on Parameters).
Chapter V. Some explicit solutions.
Chapter VI. Linear Ordinary Differential Equations.
Chapter VII. Equilibrium points and local dynamics.
Chapter VIII. Applications: population dynamics.
(Chapter IX. Numerical solution of an ODE).
The chapters between ( ) are not covered by the course and are not part of the examination material.
Exercises describe concepts analyzed in the theoretical part. Chapters are :
I. Existence and unicity.
II. ODE of the first order.
III. ODE of higher order with constant coefficients.
IV. Autonomous linear systems.
V. ODE with non-constant coefficients.
VI. Classification of equilibria.
Lectures on the blackboard with notes and "minute wooclap". Tutorial sessions (solving exercises)
The evaluation is based on a single written exam. This exam will consist of a theoretical part (a question whose answer is based on a part of the course) counting for about 1/5 of the points and exercises in the form of ODE solutions (such as illustrations from the lecture as well as exercises from the tutorial sessions) counting for about 4/5 of the points.
V. Arnol'd: Ordinary differential equations E. Hairer, S.P. Nørsett and G. Wanner: Solving Ordinary Differential Equations I. Nonstiff problems L. Pontriaguine: Ordinary differential equations G. Sansone and R. Conti: Non-linear differential equations Z. Zhang: Qualitative theory of differential equations
| Training | Block | Credits | Mandatory |
|---|---|---|---|
| Bachelor in Physics | 1 | 2 | Yes |