Courses Catalogue

Syllabus of the course: Differential Equations I

In this web page we provide the syllabus of the course Differential Equations I, offered by the Department of Physics.
The list of the courses offered during the current accademic year is available here.
The list of all courses offered by the Department of Physics is available here.

Code	Φ-211
Type	A
ECTS	7
Hours	6
Semester	Winter
Instructor	S. Sotiriadis
Program	Monday, 11:00-13:00, Room 3 Thursday, 11:00-13:00, Room 3 Friday, 11:00-13:00,Room 1
Web page	https://eclass.physics.uoc.gr/courses/PH211/
Goal of the course	The course is addressed to second year students and has as a goal to introduce the basic concepts and solution techniques of ordinary differential equations and their applications to fundamental problems in Mechanics, Electromagnetism, as well as in research fields other than physics.
Syllabus	1. Ordinary differential Equations of first order: Introduction, initial value problem. The concept of General Solution, Separation of variables, homogeneous equations, linear equations of first order, (Bernoulli and Ricatti). Exact equations and integrating factors. Simple applications (2 weeks) 2. Second order differential equations: linear equations with constant coefficients. Non homogeneous equations. Variation of parameters. Euler equations. (2 weeks) 3. Newton's equations: Application in basic problems of Mechanics. Motion under different laws of friction in a homogenous gravitational field. Harmonic Oscillation with and without friction. Forced oscillations. Motion in a one-dimentional force field. Motion under the influence of gravity. The orbits of planets. Analogues of mechanics problems in electricity. (2 weeks) 4. General study of linear differential equations: The principle of linear combination.The wronskian and its applications. The Abel type. Calculating the second solution once the first in known. Reduction of the order. Complete solution of a non-homogenous ordinary differential equation when the solution of the homogenous is known. Using Laplace transform to solve linear differential equations with constant coefficients. (3 weeks) 5. Systems of linear differential equations with constant coefficients: Methods of solutions and applications to problems of coupled oscillators and electrical circuits. Solutions using matrices. (2 weeks) 6. Linear differential equations with variable coefficients: Solutions using series. The Taylor and Frobenius series. Examples. Singular Points and Convergence of Series Solution. The theorem of Fuchs. Applications to Bessel, Legendre, Hermite and hypergeometric equations. (2 weeks)
Bibliography	Elementary Differential Equations and Boundary Value Problems, 8th Edition, by W.E. Boyce and R.C. DiPrima, John Wiley & Sons, (2005).