2023
Ordinary Differential Equations
Name: Ordinary Differential Equations
Code: MAT00913L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese, English
Regime de Frequência: Presencial
Presentation
In this UC, it is intended that the student explicitly solve ordinary, linear and non-linear differential equations, perform qualitative studies of the solution's behavior, learn techniques that guarantee the existence, uniqueness, stability of the solution and build and analyze mathematical models.
Sustainable Development Goals
Learning Goals
Acquiring explicit resolution techniques of ordinary differential equations of 1st order, linear and nonlinear, and higher order as well.
- Realize qualitative studies on the behaviour of the solution in scalar equations and planar systems.
- Know techniques that ensure the uniqueness of solution, continuous dependence and stability of the initial data of solution.
- Build and analyse mathematical models.
- Realize qualitative studies on the behaviour of the solution in scalar equations and planar systems.
- Know techniques that ensure the uniqueness of solution, continuous dependence and stability of the initial data of solution.
- Build and analyse mathematical models.
Contents
1. Ordinary Differential Equations. First order equations. Second order linear equations. Particular solution of non homogeneous equation. Homogeneous equation with constant coefficients.
2. Existence and Uniqueness of Solution. Inequalities and convergences. Picards method of sucessive approximations. Solutions extension. Uniqueness theorems. Differential inequalities and extremal solutions. Continuous dependence of initial conditions.
3. Systems of Differential Equations. Existence and uniqueness of solutions. Linear systems. Systems with constant coefficients. Asymptotic behaviour of solutions
4. Stability of Solutions. Stability of quasi-linear systems. Planar autonomous systems. Limit cycles and periodic solutions. Lyapunovs method for autonomous and nonautonomous systems. Oscillatory equations
5. Boundary value problems. Greens functions. Maximum principle. Sturm-Liouville problems. Eigenfunction expansions. Nonlinear boundary value problems
2. Existence and Uniqueness of Solution. Inequalities and convergences. Picards method of sucessive approximations. Solutions extension. Uniqueness theorems. Differential inequalities and extremal solutions. Continuous dependence of initial conditions.
3. Systems of Differential Equations. Existence and uniqueness of solutions. Linear systems. Systems with constant coefficients. Asymptotic behaviour of solutions
4. Stability of Solutions. Stability of quasi-linear systems. Planar autonomous systems. Limit cycles and periodic solutions. Lyapunovs method for autonomous and nonautonomous systems. Oscillatory equations
5. Boundary value problems. Greens functions. Maximum principle. Sturm-Liouville problems. Eigenfunction expansions. Nonlinear boundary value problems
Teaching Methods
Students will have at their disposal in Moodle all material used in classes, which are composed of an initial presentation and discussion of the topics to be studied, followed by illustrative and practical applications.
Evaluation can be made by two processes, with the possibility, in each one, to have access to material produced by students themselves:
1. Evaluation by Exam
The student will be approved if one of the exams is rated with at least 10.
2. Continuous evaluation
There will be two tests, focused on some taught chapters.
The classification of this component will be the average of the rates obtained.
The students will opt for continuous evaluation if they have at least 10 in both two tests, and, in each one, not less than eight points.
Evaluation can be made by two processes, with the possibility, in each one, to have access to material produced by students themselves:
1. Evaluation by Exam
The student will be approved if one of the exams is rated with at least 10.
2. Continuous evaluation
There will be two tests, focused on some taught chapters.
The classification of this component will be the average of the rates obtained.
The students will opt for continuous evaluation if they have at least 10 in both two tests, and, in each one, not less than eight points.
Teaching Staff
- Ana Margarida da Silva Afonso Rodrigues [responsible]
