Differential Equations
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2023/2024 - 2S
Cycles of Study/Courses
Teaching language
Portuguese
Objectives
Acquisition of basic knowledge of the theory of Differential Equations and its application to real-life problems.
Learning outcomes and competences
The students should acquire techniques which enable them:
a. to solve both classical ordinary differential equations of 1st and 2nd order and linear systems of ordinary differential equations;
b. to analyze differential equations from a qualitative point of view (equilibria, stability and phase portraits in the case of dimension 2);
c. to model (and solve) real-life problems envolving differential equations;
d. to solve classical partial differential equations (heat, wave and Laplace's equations) using separation of variables and Fourier series.
Working method
Presencial
Pre-requirements (prior knowledge) and co-requirements (common knowledge)
Real Analysis I, II and III and Linear Algebra and Analytic Geometry I and II.
Program
1. First order ordinary differential equationsLinear, separable and exact differential equations. Applications: dating through radioactive decay, population growth, mixtures, among others.
2. Theorem of existence and uniqueness of solutions 3. Systems of first order ordinary differential equationsLinear homogeneous systems with constant coefficients and nonlinear autonomous systems.
Phase portraits for systems of two autonomous differential equations.
Equilibrium points and stability.
Applications: Newton's law of cooling with variable ambient temperature and Lotka-Volterra predator-prey model.
4. Second order linear differential equationsHomogeneous linear equations. Phase portrait and graph of the solutions in the case of constant coefficientes.
Method of variation of parameters and method of reduction of order for nonhomogeneous equations.
Solutions obtained through power series expansion. Applications: motion of an object under the influence of an elastic spring, with or without friction, with or without external forces.
5. Partial differential equations
Boundary value problems. Separation of variables. Fourier series.
Heat, wave and Laplace's equations and their resolution.
Mandatory literature
Braun Martin;
Differential equations and their applications. ISBN: 0-387-90266-X
Complementary Bibliography
Hirsch Morris W.;
Differential equations, dynamical systems, and linear algebra. ISBN: 0-12-349550
Teaching methods and learning activities
Theoretical classes with exposition of the theory and illustration by examples.
Practical classes with resolution by the students of concrete problems.
keywords
Physical sciences > Mathematics > Mathematical analysis > Differential equations
Evaluation Type
Distributed evaluation with final exam
Assessment Components
designation |
Weight (%) |
Exame |
60,00 |
Teste |
40,00 |
Total: |
100,00 |
Amount of time allocated to each course unit
designation |
Time (hours) |
Estudo autónomo |
171,00 |
Frequência das aulas |
72,00 |
Total: |
243,00 |
Eligibility for exams
Not applicable.
Calculation formula of final grade
In the first call the final score will be the sum of the scores obtained in two assessments.
The first assessment, worth 8 (eight) points, will take place during the semestre, in a date to be anounced during the first lecture. The second assessment, worth 12 (twelve) points will take place during the first call of exams.
In the second call, the final score will be the obtained in an exam, worth 20 (twenty) points. The exam will be divided in two parts, corresponding to the two assessments. All the students who have not yet been approved in the course, will be given the oportunity of using their score on one of the assessments (in replacement for the resolution of the corresponding part in the exam).
Special assessment (TE, DA, ...)
Students who, by special conditions, are exempted from distributed evaluation will have an exam in the conditions described for the second call.
Classification improvement
Improvement in the classification can be obtained in the second call only and the corresponding classification will be the score obtained in the exame (worth 20 points).