Differential Equations

LEARNING OUTCOMES

The aim of the course is to understand how differential equations are used in modeling problems that arise in the specialty of Engineer and how to solve these equations. The student after the successful completion of the course will be able to:

• Understand the basic mathematical concepts and the methodology of solving first order and higher order differential equations, systems of differential equations, as well as the use of Fourier series.
• Use differential equations in modeling problems of their specialty, solve them and draw conclusions.
• Connect the mathematical methodologies taught and apply the acquired knowledge in the subject of Surveying and Geoinformatics Engineer.
• Apply all of the above in other fields of the Engineer’s specialty.

 

General Competences

• Exercise criticism and self-criticism
• Mathematical thinking and analysis
• Mathematical and analytical presentation of geometric concepts
• Search, analyze and synthesize data with the use of the necessary technology
• Autonomous work
• Production of free, creative and inductive thinking

 

SYLLABUS

The course is designed for a set of 13 weeks of lectures. The topics that will be discussed are the following:

1. Basic concepts solution of differential equation (partial and general), initial and boundary value problems.
2. First order differential equations classification and methods of solving, divisible variables, linear differential equations, homogeneous differential equations, exact differential equations, integrating factors, Bernoulli differential equation, Ricatti differential equation, applications in problems of the Engineer’s specialty.
3. Linear higher order linear differential equations with constant or variable coefficients definitions, the Wronskian, methods of solution, homogeneous solution, general solution of the linear differential equation, the method of undetermined coefficients, variation of parameters. Euler’s equations. Initial value problems and applications in engineering and electricity.
4. Solution of differential equations using power series ordinary points and singular points, existence of analytical solutions, solution to regular singular points.
5. Systems of linear differential equations, matrix method.
6. Laplace transforms definition and properties, solution of linear differential equations and systems of differential equations with constant coefficients with the Laplace transform. reduction of a differential equation in a first order system of equation.
7. Bessel equations and Legendre equations, Gamma functions, Dirac delta function.
8. Differential equations with partial derivatives, linear, boundary value problems, Separable equations. Applications in engineering.
9. Fourier series, Dirichlet type conditions, Parseval identity. Applications of the Fourier series.
10. Complex Functions and their applications.

 

STUDENT PERFORMANCE EVALUATION

During the semester students will be given problems-exercises which together with the material of the lectures will be an aid for the preparation of the final exams.

 

ATTACHED BIBLIOGRAPHY

1. Boyce W.E., Diprima R.C. and Meade D.B. (2017) – Elementary Differential Equations and Boundary Value Problems, 11th edn, Wiley.
2. Trench W.F. (2013) – Elementary Differential Equations with Boundary Value Problems, Faculty Authored and Edited Books, Trinity University.
3. Goodwine B. , 2011, Engineering Differential Equations, Springer.
4. Kalbaugh David V., 2017, Differential Equations for Engineers: The Essentials, CRC Press.
5. Kreyszig E., 2005, Advanced Engineering Mathematics, 9th edition, Wiley.
6. Glyn, J. et al., 2010, Advanced Modern Engineering Mathematics, 4th edition, Addison- Wesley Pub. Co.
7. Wylie C.R. and Barrett L.C., 1995, Advanced Engineering Mathematics, 6th edition, McGraw-Hill.
8. Zill D.G. and Cullen M.R., 2006, Advanced Engineering Mathematics, 3rd edition, Jones & Bartlett Pub.
9. Halidias, N. (2021) – Applied Mathematics for Economists and Engineers, Broken Hill. (in Greek)
10. Rassias, T. (2017) – Mathematics II, Tsotras, 2nd Edition (in Greek).