Differential Equations and Modelling

Academic year
Subject Area
Language of Instruction
Mode of Delivery
ECTS Credits
1st Cycle Studies

Recommended Prerequisites

Infinitesimal Analysis I and II; Linear Algebra and Analytic Geometry I and II.

Teaching Methods

In TP classes, the most relevant concepts and theoretical results are presented, with detail and accuracy, accompanied by illustrative examples of the theory, simple and motivating applications, and adequate historical notes. In P classes, students must solve the proposed exercises, having various degrees of difficulty, as well as be confronted with problems in the context of ODE applications. Active participation of students in class discussions, individual and team work, and correct use of  the available  office hours should be strongly encouraged by the instructor.

Learning Outcomes

The main objectives are to acquire fundamental knowledge about ordinary differential equations (ODE), including concepts and theoretical results, methods to solve them and qualitative analysis of solutions, as well as to become familiar with basic technics of mathematical modeling for solving real problems. The main competences to be developed are: ability for generalization and abstraction; ability to formulate and solve problems; design, analyze and correctly use mathematical models; ability to work in teams; critical thinking.

Work Placement(s)



I- ODE and mathematical modeling: basic notions, construction/validation of models, classic examples.

II- First-order ODE: linear equation, separable variables equation, existence/uniqueness of solutions, construction and analysis of models for real situations

III- Linear ODE of order n: definition and classification, existence/uniqueness of solutions, homogeneous equations, Lagrange method, annihilator polynomial method, construction and analysis of models for real situations.

IV- Systems of 1st order linear ODE: basic definitions and matrix form, existence and uniqueness theorem, matrix exponential, homogeneous systems, finding fundamental systems of solutions, variation of parameter’s method, construction and analysis of models for real situations.

V- Two-dimensional qualitative theory: equilibrium solutions, stability, complete characterization of stability for linear systems, linearization, phase space, orbits, phase portrait, analysis of models.

Head Lecturer(s)

Maria de Fátima da Silva Leite

Assessment Methods

Exam (100%) or Midterm exam (70%) + test (15%) + Project (15%): 100.0%


 M. Braun, Differential equations and their applications. 4rd ed., Springer Verlag, 1993.


D. Burghes, B. Morrie, Modeling with differential equations. John Wiley & Sons, 1981.


D. Figueiredo, A Neves, Equações Diferenciais Aplicadas. R. Janeiro, IMPA, 2010.


F. Leite, J. Petronilho, Notas de equações diferenciais e modelação (em construção).


G. Simmons, Differential Equations with Applications and Historical Notes. McGraw Hill, 2003.


D. G. Zill, Equações diferenciais com aplicação em modelagem. Thomson, 2003.