Bayburt University Lifelong Learning Programme

Description of Individual Course Units

Course Unit Code	Course Unit Title	Type of Course Unit	Year of Study	Semester	Number of ECTS Credits
G209B2	Differential Equations	Compulsory	2	3	4

Level of Course Unit

First Cycle

Objectives of the Course

This course aims to provide students with general knowledge on formulating problems that arises in applied sciences as mathematical models, solving such models through analytical, qualitative and numerical methods, as well asinterpreting solutions within the concept of physical problem at hand.

Name of Lecturer(s)

Dr. Öğr. Üyesi Hüseyin KÖKSAL

Learning Outcomes

1	Formulate mathematical models for a variety of problems
2	Solve the model using analytical, qualitative and partically some numerical methods
3	Interprate the solution within the concept of the phenomenon being modeled
4	Obtain solution for models studied within the scope of the course
5	Will learn the physical applications of differential equations.

Mode of Delivery

Normal Education

Prerequisites and co-requisities

None

Recommended Optional Programme Components

None

Course Contents

Differential equations and basic concepts. Differential equations as mathematical model (Ordinary differential equations, order and degree of differential equations. Derivation of differential equations.) General, particular and singular solutions of the differential equations. Existence and uniqueness theorems. Direction fields and solution curves. Separable, homogenous, exact differential equations and transforming to exact differential equation by using integrating factor. Linear differential equations, Bernoulli differential equation and applications of the first order differential equations(Population model, acceleration-velocity model, temperature problems). Change of variables. Reducible differential equations (single variable and non-linear differential equations). General solution of nth order linear differential equations (linearly independent solutions, super position principle for the homogeneous equations, particular and general solutions). General solution of nth order constant coefficient homogenous differential equations. Solutions of the constant coefficient non-homogenous equations. (Undetermined coefficients, change of parameters). Initial Value Problems (IVP) and Boundary Value Problems (BVP) (Eigenvalues and eigenfunctions for boundary value problems. Physical applications, mechanical vibrations, electrical circuits). Variable coefficient homogenous and non-homogenous differential equations (Cauchy-Euler, Legendre differential equations). Reduction of order. Power series solutions of differential equations around ordinary points.Laplace and inverse Laplace transformations. Solutions of constant and variable coefficient boundary value problems and differential equations containing Dirac-Delta function and transformation functions by using Laplace transformations. System of differential equations. Transformation of higher order differential equation to the system of first order differential equations. Solutions of the homogenous differential equations using eigenvalues and eigenvectors. Solutions of non-homogeneous constant coefficient system of differential equations. Application of the Laplace transformation to system of differential equations. Numerical solutions of differential equations (Euler and Runge-Kutta methods)

Weekly Detailed Course Contents

Week	Theoretical	Practice	Laboratory
1	Differential equations and basic concepts. Differential equations as mathematical model.
2	General, particular and singular solutions of the differential equations. Existence and uniqueness theorems. Direction fields and solution curves.
3	Differential equations that can be separated into their variables and transformed into homogeneous, exact and exact forms.
4	Linear differential equations, Bernoulli differential equation and applications of the first order differential equations(Population model, acceleration-velocity model, temperature problems)
5	Change of variables. Reducible differential equations (single variable and non-linear differential equations).
6	General solution of nth order linear differential equations (linearly independent solutions, super position principle for the homogeneous equations, particular and general solutions). General solution of nth order constant coefficient homogenous differential equations.
7	Non-homogeneous equations with constant coefficients and solution methods. (Uncertain coefficients method, Variation of parameters method).
8	Midterm exam
9	Initial and boundary value problems. (Eigenvalues, eigenfunctions for boundary value problems. Physical applications, mechanical vibrations, Electrical circuits).
10	Homogeneous and non-homogeneous differential equations with variable coefficients (Cauchy-Euler, Legendre differential equations). Demotion method.
11	Solution of differential equations with the help of series around the common point.
12	Laplace and inverse Laplace transforms.
13	Initial value problems with constant and variable coefficients and solutions of differential equations including Delta-Dirac and translation functions using the Laplace method. moment, center of gravity and work.
14	Systems of differential equations. Conversion of higher order differential equations to first order system. Solution of homogeneous systems of differential equations using eigenvalue and eigenvector methods. Solutions of non-homogeneous systems of constant coefficient differential equations.
15	Application of Laplace transforms to systems of differential equations. Numerical solution methods for differential equations.