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 as interpreting solutions within the concept of physical problem at hand.

Learning Outcomes

CTPO

TOA

Upon successful completion of the course, the students will be able to :

LO - 1 :

formulate mathematical models for a variety of problems

1,2

1,

LO - 2 :

solve the model using analytical, qualitative and partically some numerical methods,
solve the model using analytical, qualitative and partically some numerical methods

1,2

1,

LO - 3 :

interprate the solution within the concept of the phenomenon being modelled

1,2

1,

LO - 4 :

obtain solution for models studied within the scope of the course

1,2

1,

CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), LO : Learning Outcome

Contents of the Course

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).

Course Syllabus

Week

Subject

Related Notes / Files

Week 1

Differential equations and basic concepts. Differential equations as mathematical model (Ordinary differential equations, order and degree of differential equations. Derivation of differential equations.)

Week 2

General, particular and singular solutions of the differential equations. Existence and uniqueness theorem.

Week 3

Separable, homogenous differential equations.

Week 4

Exact differential equations and transforming to exact differential equation by using integrating factor.

Week 5

Linear Differential Equations, Bernoulli Differential Equation.

General solution of nth order linear differential equations (linearly independent solutions, super position principle for the homogeneous equations, particular and general solutions).

Week 8

General solution of nth order constant coefficient homogenous differential equations.

Week 9

mid-term exam

Week 10

Solutions of the constant coefficient non-homogenous equations. (Undetermined coefficients, change of parameters)

Week 11

Cauchy Euler Equations.

Week 12

Power series solutions of differential equations around ordinary points.

Week 13

Laplace and inverse Laplace transformations.

Week 14

Solutions of differential equations by using Laplace transformations.

Week 15

System of differential equations. Solutions of the homogenous differential equations using eigenvalues and eigenvectors.

Week 16

final exam

Textbook / Material

1

Edwards, C.H., Penney, D.E. (Çeviri Ed. AKIN, Ö). 2006; Diferensiyel Denklemler ve Sınır Değer Problemleri (Bölüm 1-7), Palme Yayıncılık, Ankara.

Recommended Reading

1

COŞKUN, H. 2002; Diferansiyel Denklemler, KTÜ Yayınları, Trabzon.

2

BAŞARIR, M., TUNCER, E.S. 2003; Çözümlü Problemlerle Diferansiyel Denklemler, Değişim Yayınları, İstanbul.