EPOKA UNIVERSITY
FACULTY OF ARCHITECTURE AND ENGINEERING
DEPARTMENT OF CIVIL ENGINEERING
COURSE SYLLABUS
2025-2026 ACADEMIC YEAR
COURSE INFORMATIONCourse Title: DIFFERENTIAL EQUATIONS |
| Code | Course Type | Regular Semester | Theory | Practice | Lab | Credits | ECTS |
|---|---|---|---|---|---|---|---|
| MTH 201 | A | 3 | 3 | 0 | 0 | 3 | 5 |
| Academic staff member responsible for the design of the course syllabus (name, surname, academic title/scientific degree, email address and signature) | Dr. Valmir Bame vbame@epoka.edu.al |
| Main Course Lecturer (name, surname, academic title/scientific degree, email address and signature) and Office Hours: | M.Sc. Bredli Plaku bplaku@epoka.edu.al , By appointment. |
| Second Course Lecturer(s) (name, surname, academic title/scientific degree, email address and signature) and Office Hours: | NA |
| Language: | English |
| Compulsory/Elective: | Compulsory |
| Study program: (the study for which this course is offered) | Bachelor in Civil Engineering (3 years) |
| Classroom and Meeting Time: | |
| Teaching Assistant(s) and Office Hours: | NA |
| Code of Ethics: |
Code of Ethics of EPOKA University Regulation of EPOKA University "On Student Discipline" |
| Attendance Requirement: | 60% |
| Course Description: | First-order differential equations, second-order linear equations, change of parameters, homogeneous and non-homogeneous equations, series solutions, Laplace transform, systems of first-order linear equations, boundary value problems, Fourier series. |
| Course Objectives: | This course equips students with a strong grasp of ordinary differential equations (ODEs) while highlighting their significance in civil engineering. Students will learn to identify ODE types and apply solving techniques. Through practical applications in civil engineering, including structural analysis, fluid dynamics, and environmental modelling, students will see how ODEs play a pivotal role in addressing real-world challenges within the field. |
|
BASIC CONCEPTS OF THE COURSE
|
| 1 | Differential Equation (DE): An equation involving an unknown function and its derivatives. |
| 2 | Order of a DE: The highest derivative present in the equation. |
| 3 | Degree of a DE: The power of the highest derivative after the equation is made polynomial in derivatives. |
| 4 | Ordinary vs. Partial DEs: ODEs involve functions of a single independent variable; PDEs involve functions of several variables. |
| 5 | Linearity: A DE is linear if the unknown function and its derivatives appear to the first power and are not multiplied together. |
| 6 | General Solution: A family of solutions containing arbitrary constants equal to the order of the equation. |
| 7 | Particular Solution: A specific solution obtained by fixing constants, often satisfying initial or boundary conditions. |
| 8 | Initial Value Problem (IVP): A DE together with conditions that specify the value of the solution and its derivatives at a point. |
| 9 | Boundary Value Problem (BVP): A DE together with conditions specified at more than one point. |
| 10 | Direction (Slope) Field: A graphical representation of solution behaviour by plotting tangent slopes to the solution curves at sample points. |
|
COURSE OUTLINE
|
| Week | Topics |
| 1 | Introduces differential equations (order; ODE vs PDE; linear vs nonlinear), the ideas of general vs particular solutions, initial conditions/IVPs, and intervals of validity in context. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 1–8. |
| 2 | Qualitative analysis via direction fields (slope fields): how to sketch them, read solution behaviour without solving, and link to basic modelling narratives. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 9–22. |
| 3 | First-order solution methods: linear, separable, and exact equations with step-by-step procedures and typical pitfalls. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 22–62. |
| 4 | Further first-order topics: Bernoulli, useful substitutions, intervals of validity and existence/uniqueness ideas; applied modelling (mixing, population, falling bodies with drag); equilibrium solutions and Euler’s method. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 63–120. |
| 5 | Second-order homogeneous linear ODEs with constant coefficients: characteristic equation with real, complex, and repeated roots; reduction of order, fundamental sets, and the Wronskian. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 121–161. |
| 6 | Second-order nonhomogeneous equations: complementary/particular decomposition, undetermined coefficients, variation of parameters, and mechanical vibrations (free/forced, damping). Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 162–212. |
| 7 | Midterm examination covering Weeks 1–6. |
| 8 | Laplace transforms: definition, basic transform tables and properties, and inverse Laplace techniques. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 213–235, 277–278. |
| 9 | Laplace methods for initial value problems including step functions and non-constant coefficients. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 236–268, 277–278. |
| 10 | Systems of differential equations: linear-algebra review (matrices, eigenvalues/eigenvectors), writing systems in matrix form, solution structure, phase plane concepts, and solution types (real/complex/repeated eigenvalues), with brief notes on nonhomogeneous systems and Laplace for systems. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 279–366. |
| 11 | Boundary value problems and eigenvalue/eigenfunction formulation (Sturm–Liouville flavour) to prepare for separation of variables. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 427–461. |
| 12 | Series solutions about ordinary points. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 367–397. |
| 13 | Fourier analysis for PDEs: periodic/orthogonal functions, sine and cosine series, full Fourier series, and basic convergence facts. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 462–496. |
| 14 | Introduction to partial differential equations: heat and wave equations, terminology, and separation of variables with standard boundary conditions. Literature: Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. pp. 497–555. |
| Prerequisite(s): | Calculus I and Calculus II |
| Textbook(s): | Boyce WE, DiPrima RC, Meade DB. Elementary Differential Equations and Boundary Value Problems. 12th ed. Hoboken (NJ): Wiley; 2021. 640 p. ISBN: 978-1119777694. |
| Additional Literature: | Dawkins P. Paul’s Online Math Notes: Differential Equations. Beaumont (TX): Lamar University; 2022. Available from: https://tutorial.math.lamar.edu/ |
| Laboratory Work: | |
| Computer Usage: | |
| Others: | No |
|
COURSE LEARNING OUTCOMES
|
| 1 | Classify differential equations by order, linearity, and type (ODE vs PDE) and explain their role in mathematical modelling. |
| 2 | Construct and interpret direction fields to analyse qualitative behaviour of first-order differential equations. |
| 3 | Solve first-order differential equations using standard techniques such as separation of variables, integrating factors, exact equations, and special forms (Bernoulli, Euler, homogeneous). |
| 4 | Apply differential equations to model physical, biological, and engineering systems (e.g., population dynamics, mixing, mechanical systems). |
| 5 | Solve second-order linear homogeneous and nonhomogeneous differential equations with constant coefficients using the characteristic equation, undetermined coefficients, and variation of parameters. |
| 6 | Analyse mechanical vibration problems, including free and forced oscillations with and without damping, using second-order models. |
| 7 | Utilise Laplace transforms to solve initial value problems involving discontinuous forcing functions and general inputs. |
| 8 | Solve and interpret systems of linear differential equations using matrix methods, eigenvalues/eigenvectors, and phase plane analysis. |
| 9 | Apply series methods and Fourier analysis to represent and solve problems involving boundary value problems and partial differential equations. |
| 10 | Recognise the importance of differential equations in civil engineering by relating solution methods to applications such as structural vibrations, groundwater flow, and heat transfer in materials. |
|
COURSE CONTRIBUTION TO... PROGRAM COMPETENCIES
(Blank : no contribution, 1: least contribution ... 5: highest contribution) |
| No | Program Competencies | Cont. |
| Bachelor in Civil Engineering (3 years) Program | ||
| 1 | an ability to apply knowledge of mathematics, science, and engineering | 5 |
| 2 | an ability to design a system, component, or process to meet desired needs | 5 |
| 3 | an ability to function on multidisciplinary teams | 5 |
| 4 | an ability to identify, formulate, and solve engineering problems | 5 |
| 5 | an understanding of professional and ethical responsibility | 5 |
| 6 | an ability to communicate effectively | 4 |
| 7 | the broad education necessary to understand the impact of engineering solutions in a global and societal context | 4 |
| 8 | a recognition of the need for, and an ability to engage in life long learning | 4 |
| 9 | a knowledge of contemporary issues | 3 |
| 10 | an ability to use the techniques, skills, and modern engineering tools necessary for engineering practice | 3 |
| 11 | skills in project management and recognition of international standards and methodologies | 3 |
|
COURSE EVALUATION METHOD
|
| Method | Quantity | Percentage |
| Midterm Exam(s) |
1
|
35
|
| Presentation |
1
|
4
|
| Quiz |
2
|
10
|
| Case Study |
1
|
6
|
| Final Exam |
1
|
35
|
| Total Percent: | 100% |
|
ECTS (ALLOCATED BASED ON STUDENT WORKLOAD)
|
| Activities | Quantity | Duration(Hours) | Total Workload(Hours) |
| Course Duration (Including the exam week: 16x Total course hours) | 16 | 3 | 48 |
| Hours for off-the-classroom study (Pre-study, practice) | 14 | 2 | 28 |
| Mid-terms | 1 | 16 | 16 |
| Assignments | 1 | 7 | 7 |
| Final examination | 1 | 16 | 16 |
| Other | 2 | 5 | 10 |
|
Total Work Load:
|
125 | ||
|
Total Work Load/25(h):
|
5 | ||
|
ECTS Credit of the Course:
|
5 | ||
|
CONCLUDING REMARKS BY THE COURSE LECTURER
|
|
The course aims to strengthen both theoretical understanding and practical problem-solving in differential equations, with applications relevant to civil engineering. Students are encouraged to study continuously and make use of support resources when needed. The lecturer commits to fairness and professionalism in teaching and assessment, while students are expected to uphold academic integrity and comply with the University’s Code of Ethics. |