ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Hours of tutorials: 1 Expository Class: 33 Interactive Classroom: 17 Total: 51
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Applied Mathematics
Areas: Applied Mathematics
Center Higher Technical Engineering School
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
1. To know and handle the terminology related to ordinary differential equations and partial differential equations.
2. To identify how engineering and applied science problems are translated into mathematical models involving differential equations. To analyze the adequacy and limitations of these models in representing real phenomena.
3. To know and correctly apply the analytical methods and numerical techniques included in the course descriptors for their resolution.
4. To classify and solve the most common ordinary differential equations, especially the case of linear equations. To master analytical techniques such as separation of variables, integrating factor, and methods of variation of parameters and undetermined coefficients.
5. To apply the Laplace transform to solve differential equations, and especially to understand those with a discontinuous or impulsive second member. To apply the Laplace transform to solve systems of linear differential equations.
6. To recognize the limitations of analytical methods and the importance of numerical methods for solving complex differential equations and to study the most basic ones.
7. To use MATLAB for solving differential equations. To validate the numerical results obtained and compare them with theoretical solutions where possible, analyzing the accuracy of the approximation.
Topic 1. Introduction to Ordinary Differential Equations (ODEs)
Motivation. Basic terminology: order, type, and linearity. General and particular solutions. Singular solutions. Existence and uniqueness of solutions for a first-order initial value problem. Some engineering problems leading to ODEs.
Topic 2. First-order ODEs
Equations with separable variables. Exact equations. Integrating factor. Linear equations. Homogeneous equations. Applications of first-order ODEs.
Topic 3. Introduction to Numerical Solutions of ODEs
Motivation. Generalities. Numerical solution of a first-order initial value problem. Euler's method. Second-order Runge-Kutta methods. Applications.
Topic 4. Higher-order Linear ODEs
Second-order linear equations. Homogeneous linear equations with constant coefficients. General solution. Non-homogeneous linear equations with constant coefficients. Method of undetermined coefficients and method of variation of parameters. Higher-order linear equations. Applications. Numerical solution of higher-order differential equations.
Topic 5. Solving Systems of Linear ODEs. Laplace Transform
Definition of the Laplace transform. Calculation and properties of the Laplace transform. Inverse Laplace transform. Application to solving systems of linear differential equations. Applications in chemical engineering.
Topic 6. Introduction to Partial Differential Equations (PDEs)
Definition of a PDE. Order and solution of a PDE. Second-order linear PDEs. Example: the heat equation. Method of separation of variables.
MATLAB PRACTICAL SESSIONS
These practical sessions will take place during interactive laboratory classes in the computer lab. Students will individually solve problems and case studies related to Chemical Engineering using the MATLAB software package.
All the topics described will be developed and applied in the lecture sessions as well as in the interactive seminar and MATLAB practical classes.
Enrolled students will have access to materials related to the course content on the course website hosted on the virtual campus.
BASIC BIBLIOGRAPHY:
• NAGLE, R. Kent, SAFF, Edward B., SNIDER, Arthur David, 2005. Ecuaciones diferenciales y problemas con valores en la frontera. 4ª ed. México: Pearson Education. ISBN 978-968-444-483-6
• NAGLE, R. Kent; SAFF, Edward B.; SNIDER, Arthur David, 2018. Fundamentals of Differential Equations. 9ª ed. Harlow: Pearson. ISBN 978-1-292-24099-2.
Available on line (PreLo):
• NAGLE, R. Kent, SAFF, Edward B., SNIDER, Arthur David, 2005. Ecuaciones diferenciales y problemas con valores en la frontera. Ebook. 4ª ed. Naucalpan de Juárez: Pearson Educación. ISBN 9786074426175. Dispoñible en: https://www-ingebook-com.ezbusc.usc.gal/ib/NPcd/IB_BooksVis?cod_primari…. [Consulta: 13-05-2025]
COMPLEMENTARY BIBLIOGRAPHY:
• BOYCE, William E., DIPRIMA, Richard C., 2010. Elementary Differential Equations and Boundary Value Problems. 9ª ed. New York: Wiley. ISBN 978-0-470-39873-9
• CUTLIP, Michael B., SHACHAM, Mordechai, 2000. Problem solving in chemical engineering with numerical methods. New Jersey: Prentice Hall International Series in the Physical and Chemical Engineering Sciences. ISBN 0-13-862566-2
• SIMMONS, George F., 2002. Ecuaciones diferenciales con aplicaciones y notas históricas. 2ª ed. Madrid: McGraw-Hill. ISBN 84-481-0045-X
• ZILL, Dennis G., CULLEN, Michael R., 2008. Matemáticas avanzadas para ingeniería I: ecuaciones diferenciales. 3ª ed. México: McGraw-Hill. ISBN 9789701065143
Knowledge or Content
Con18: Knowledge in basic and technological subjects that enables them to learn new methods and theories and provides versatility to adapt to new situations.
Skills or Abilities
H/D05: Ability to apply knowledge in practice.
Competencies
Comp03: Ability to solve mathematical problems that may arise in engineering. Aptitude to apply knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; differential equations and partial differential equations; numerical methods; numerical algorithmics; statistics and optimization.
Teaching is organized into lectures and interactive activities (seminars, computer lab sessions, and tutorials), combining active methodologies and digital resources to support the acquisition of competencies:
A) Lectures (large group): Explanation of theoretical content, problems, and general examples related to Chemical Engineering, supported by slides available on the virtual campus. The aim is to provide a solid conceptual foundation. The instructor will use the references listed in the “Basic Bibliography” section.
B) Interactive seminar sessions (small group): Guided resolution of practical exercises included in the problem sets for each topic, encouraging active participation and critical thinking. Both the problem sets and their solutions are published on the course’s virtual platform.
C) Interactive computer lab sessions (small group): Individual application of acquired knowledge through problem-solving using MATLAB. Students will complete a series of guided exercises (previously provided via the virtual campus) in which this software is used to illustrate examples or solve problems introduced in lectures or seminars. These sessions help develop technical and problem-solving skills.
D) Tutorial session (very small group): Scheduled tutorials provided by the instructor, amounting to one hour per student during the semester. These sessions are intended to resolve doubts and review continuous assessment tasks, supporting personalized learning.
E) Virtual resources: Enrolled students will have access to materials related to the content covered in the various sessions (slides, problem sets, lab instructions, and solutions) on the course website hosted on the virtual campus, to facilitate course follow-up. Additionally, supplementary material will be provided, illustrating real-world applications of the course content in the field of Chemical Engineering, with the aim of reinforcing the connection between theory and its practical relevance in professional contexts. This supplementary material is for educational purposes only and will not be assessed; this will be clearly indicated when applicable.
F) Tutorials: The instructor will offer a weekly tutorial schedule, published on the course website. These sessions are intended to address specific questions from students who have previously engaged with the course content.
The assessment system follows the general criteria established in the Degree in Chemical Engineering curriculum. Each student’s final grade will be based on continuous assessment and a final written exam, both applicable in the two official examination periods. These are the only assessment activities planned for the course, and their completion is mandatory for all students.
1) Mandatory Assessment Activities
• Final written exam (ET): In each examination period, students will take a final written exam on the official dates set by the School. A minimum score of 3 out of 7 is required to pass the course. This exam assesses both theoretical and practical knowledge of the subject.
• Midterm written test (EC): A written test held around the middle of the semester, focusing on the theoretical and practical content covered up to that point.
• MATLAB practical test (EP): Conducted in the computer lab after the completion of the practical sessions. It evaluates the practical application of knowledge using MATLAB.
2) Weighting of Each Assessment Activity in the Final Grade
The weighting of each activity in the student's final mark is summarised below:
- Grade for the final written examination (ET): 70% (7 marks out of 10).
- Grade for Matlab practical test (EP): 15% (1.5 marks out of 10).
- Intermediate continuous assessment test (EC) mark: 15% (1.5 out of 10).
In either of the two examination periods, the final grade is calculated as: Final Grade (CG) = ET + EP + EC
The results of the continuous assessment will be communicated to students before the final exam and will be retained for the second examination period if necessary.
A student will receive a grade of “Absent” if they do not complete any assessable academic activity and do not attend either of the official exams.
4) Repeating Students
All repeating students must follow the same assessment system as regular students, except for the computer lab sessions: a passing grade (equal to or greater than 0.75) in the MATLAB practical test from the 2024–25 academic year will be carried over to the 2025–26 academic year.
5) Competency Assessment
Each of the assessment activities evaluates the knowledge, skills, and competencies described in the “Competencies” section, specifically: Con18, H/D05, Comp03.
6) Academic Integrity
In cases of fraudulent conduct during exercises or tests, the provisions of the Regulations on the Assessment of Academic Performance and Review of Grades will apply.
7) Attendance and Regulations
According to Article 1 of the Regulations on Class Attendance in Official Undergraduate and Master’s Programs at USC (approved in November 2024), class attendance is considered a right and duty of students, and each course syllabus defines the mandatory learning activities.
In this course, attendance is not mandatory to pass and will not be considered in the assessment, in accordance with option d) of the aforementioned article.
Nevertheless, active participation in face-to-face sessions is strongly encouraged, as it enhances understanding of the content, supports the development of individual and group competencies, and contributes to a more enriching and participatory learning experience.
Theoretical teaching: 33h
Interactive teaching seminar: 7h
Interactive teaching laboratory/computer lab: 10h
Small group tutoring: 1h
Exams and review: 4h
Personal work of the student: 95h
- To have taken and passed the Mathematics course taught in the first semester.
- To study the subject regularly.
- To attend and actively participate in classes.
- To attend tutorials to discuss, comment on, or resolve doubts.
The Virtual Campus will be used.
The language of instruction for the classes is the one listed on the official USC website for this course.
Maria Dolores Gomez Pedreira
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813186
- mdolores.gomez [at] usc.es
- Category
- Professor: University Lecturer
| Tuesday | |||
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| 10:00-11:00 | Grupo /CLE_01 | Spanish | Classroom A2 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Classroom A2 |
| Thursday | |||
| 10:00-11:00 | Grupo /CLIS_02 | Spanish | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /CLIL_02 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /TI-ECTS01 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /TI-ECTS04 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /TI-ECTS07 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /CLIL_01 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /CLE_01 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /TI-ECTS03 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /TI-ECTS06 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /CLIS_02 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /CLIL_03 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /TI-ECTS02 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /TI-ECTS05 | Classroom A2 |
| 05.21.2026 09:15-14:00 | Grupo /CLIS_01 | Classroom A2 |
| 06.22.2026 09:30-14:00 | Grupo /CLIL_03 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /TI-ECTS02 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /TI-ECTS05 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /CLIS_01 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /CLIL_02 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /TI-ECTS01 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /TI-ECTS04 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /TI-ECTS07 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /CLIL_01 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /CLE_01 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /TI-ECTS03 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /TI-ECTS06 | Classroom A1 |
| 06.22.2026 09:30-14:00 | Grupo /CLIS_02 | Classroom A1 |