Differential Equations

Code:

M2035

Acronym:

M2035

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 2S

Active?	Yes
Web Page:	https://moodle2324.up.pt/course/view.php?id=2254
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:B	0	Official Study Plan	3	-	9	72	243
L:CC	0	study plan from 2021/22	2	-	9	72	243
			3
L:F	0	Official Study Plan	2	-	9	72	243
			3
L:G	0	study plan from 2017/18	2	-	9	72	243
			3
L:M	0	Official Study Plan	2	-	9	72	243
L:MA	0	Official Study Plan	2	-	9	72	243
L:Q	0	study plan from 2016/17	3	-	9	72	243

Teaching language

Portuguese

Objectives

Acquisition of basic knowledge of the theory of Differential Equations and Fourier Analysis.

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 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 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 

3. Systems of first order ordinary differential equations
Linear homogeneous systems with constant coefficients.

4. Second order linear differential equations
Homogeneous 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

With the final exam of 20 points.