27009 - Ordinary Differential Equations
Compulsory
5.1. Methodological overview
The course contains theoretical classes, including examples, and exercise sessions in small groups. Interaction between the teacher and the student is encouraged. Exercises proposed to the students will be presented to the group. Students will be attended by the teacher at office hours. Students will be required to from a group and to study a concrete problem and prepare a written report, to be discussed in the classroom.
5.3. Syllabus
I.- Linear systems: constant coefficients
1. Linear differential equations with constant coefficients
First-order homogeneous equation
First-order nonhomogeneous equation
Second order equations
2. Homogeneous linear systems: obtaining solutions
Eigenvectors and eigensolutions
Generalized eigenvectors
Generalized eigensolutions
3. Exponential Matrix
Convergence
Exponential matrix definition and first properties
Exponential matrix via generalized eigensolutions
Differential of the exponential matrix
4. Linear systems
Solution of homogeneous system
Solution of a nonhomogeneous system
Higher-order differential equations
5. Qualitative theory
Notion of stability
Stability and spectrum
Phase portrait. Classification of 2-d systems.
6. Laplace transform
Laplace transform defined
Calculus of Laplace transform
Calculus of inverse Laplace transform
Solution of initial value problems
Stability
II.- Linear systems: general case
7. Linear equations
Homogeneous equations
Nonhomogeneous equations
Grönwall inequality
8. Linear systems
Existence and uniqueness of solutions (homogeneous system)
Superposition principle. Resolvent matrix
Nonhomogeneous equations
Higher-order equations
Stability*
9. Periodic systems*
Periodic solutions
Structure of the solution
Stability and resonance
III.- Nonlinear systems
10. Autonomous equations
Some examples and properties
Existence and uniqueness. Asymptotes
Qualitative analysis
11. Nonautonomous equations
Exact equations
Integrating factors
Other methods (separable, homogeneous,...)
12. Existence and uniqueness
Lipchitz functions
Existence and uniqueness: Picard theorem
Maximal solution
13. Numerical methods
Euler methods and Taylor method
Convergence
Runge-Kutta method
Multistep methods*
14. Regularity of the general solution
Continuous dependence
Smooth dependence.
The variational equation
Trivialization*
15. Qualitative theory
Autonomous systems
Stability of equilibria: linearization method
Stability of equilibria: Lyapunov functions
Phase diagram