Wintersemester

Differential and Difference Equations

One-semester course of lectures and practical trainings.

The course taught for the 2^nd-year students of Bachelor program “Applied Mathematics”.

The classical topics of the theory of ordinary differential equations and systems are considered. The theory of linear systems and methods of qualitative study of linear differential equations is given in details.

Course Objective: exploration of fundamental methods of differential equations’ theory as a math tool for determined occurrences, acquirement of essential methods for differential equations’ study, observation of different integration techniques for equations and systems of various kinds.

Course Tasks:

-     Formation of practical skills for solution and research of ordinary differential equations of basic types.

-     Acquaintance with solution methods for integrable types of differential equations, techniques of quality research and implementation of differential equations in math modeling of dynamic processes.

-     Acquirement of skills for practical tasks’ modeling by differential equations.

Estimated Results:

Students completed this course have to

Know:

-       main tasks of differential equations’ theory;

-       essential methods for integration of differential equations;

-       basic elements of stability theory of differential equation system

Be able to:

-       solve first-order differential equations of basic types; 

-       set and solve Cauchy problem;

-       solve linear equations and systems with constants coefficients;

-       solve boundary value problems;

-       study solution stability and create path on phase plane.

Understand:

  probatory technique for main theorems of differential equation theory.

Content

1. Introduction

     Preliminaries

     Sample Application of Differential Equations

2. First Order Ordinary Differential Equations (ODE)

-              Separable Equations

-              Exact Differential Equations

-              Integrating Factors

-              Linear First Order Equations

-              Substitutions

-              Bernoulli Equation

-              Homogeneous Equations

-              Substitution to Reduce Second Order Equations to First Order

3. Applications and Examples of First Order ODE

-              Orthogonal Trajectories

-              Exponential Growth and Decay

-              Population Growth

-              Predator-Prey Models

-              Newton’s Law of Cooling

-              Water Tanks

-              Motion of Objects Falling Under Gravity with Air Resistance

-                   Escape Velocity

-              Planetary Motion

-              Particle Moving on a Curve

4. Linear Differential Equations

-              Homogeneous Linear Equations

-              Linear Differential Equations with Constant Coefficients

-              Nonhomogeneous Linear Equations

-              Undetermined Coefficients

-              Variation of Parameters

-              Substitutions: Euler’s Equation

5. Linear Systems

-              Preliminaries

-              Computing e T

-              The 2 × 2 Case in Detail

-              The Non-Homogeneous Case

6. Existence and Uniqueness Theorems

-              Picard’s Method

-              Existence and Uniqueness Theorem for First Order ODE’s

-              Existence and Uniqueness Theorem for Linear First Order ODE’s

-              Existence and Uniqueness Theorem for Linear Systems

7. Numerical Approximations

-              Euler’s Method

-              Error Bounds

-              Improved Euler’s Method

-              Runge-Kutta Methods

Literatur