Differential Equations

Code:

M2035

Acronym:

M2035

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=4746
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:M	78	Official Study Plan	2	-	9	84	243

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 and II and Linear Algebra and Analytic Geometry I and II.

Program

1. First order ordinary differential equations
Linear, separable and exact differential equations. Applications: dating through radioactive decay, population growth, mixtures, among others.

2. Theorem of existence and uniqueness of solutions
Statement and proof of the theorem using Banach's fixed point theorem. 

3. Systems of first order ordinary differential equations
Linear homogeneous systems with constant coefficients and nonlinear autonomous systems. Phase portraits. 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 equations
Homogeneous equations. 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 without final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	156,00
Frequência das aulas	84,00
Trabalho escrito	3,00
Total:	243,00

Eligibility for exams

Students should be present in all classes used for evaluation (to settle in the lectures of the first week).

Calculation formula of final grade

First call: the score will be the weighted average of the scores obtained in three assessments, each one worth 20 points, with weights 35%, 35% and 30%, respectively. In order to be approved the student must obtain a minimum score of 5 points in each of the first two assessment and of 4 points in the third assessment.

Second call: the score will be the one obtained in an exam, divided in three parts, each one corresponding to the contents of an assessment. The requirement of a minimum score of 5 points in each of the first two parts and of 4 points in the third part will still hold. 
Students who have not been previously approved in the UC, and only these students, will be given the possibility of replacing one of the parts of the exame by the score obtained in the corresponding assessment. 

Exception applicable to both calls: a student with a score greater than or equal to 18 will have a complementary assessment (in a date to be settled), otherwise his final score will be 18.

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.