BSc: Differential Equations.f23

Differential Equations

• Course name: Differential Equations
• Code discipline: CSE205
• Subject area: Math

Short Description

This course is an introduction to ordinary differential equations(ODEs) and their applications. Topics covered include first order ODEs, second order linear ODEs, Laypunov’s stability theory and numerical methods.The course will also introduce students to systems of linear equations and eigenvalue problems.

Course Topics

Course Sections and Topics

Section	Topics within the section
Differential equations of the first order	Introduction into differential equations. Origins and examples of the differential equations. A geometrical interpretation of the differential equations of the first order. A field of directions and solutions of the differential equations as trajectories Theorem about existence of the solution of the differential equations. Proof of the theorem. Equations with separable variables and linear equations of the first order. Homogeneous equations and exact equations and integration factors.
Differential equations of the second order	Linear equations of the second order. Phase portraits, trajectories and conservation laws. Singular points of the second order equations. Non-homogeneous equations and method of undetermined coefficients. Resonances. Variations parameters for the second order equations. Boundary value problems for the second-order equations and Green's function. Applications of Laplace transform.
Nonlinear equations and Lyapunov's stability	Nonlinear equations of the second order. Conservation laws and trajectories. Lyapunov's stability. Lyapunov's function and Lyapunov stability theorems. Chetaev's instability theorem and examples of chaotic systems. Partial differential equations of the first order and method of characteristics.
Systems of the differential equations	Linear systems of differential equations of the first order and matrix of fundamental solutions. Method of variations parameters for the non-homogeneous linear systems.
Numerical methods	Euler's method. Runge-Kutta method. Stability and accuracy of numerical methods.

Intended Learning Outcomes (ILOs)

Course objectives

Upon completion of this course, students should be able to:

• Realize conditions of existence for the equations of the first order and solve first-order ordinary differential equations using various techniques such as separation of variables, integration factors.
• Solve second-order linear differential equations with constant coefficients using techniques such as the characteristic equation and the method of undetermined coefficients and applications of Laplace transform for the linear equations.
• Define the resonant conditions for the linear and nonlinear equations of the second order equation.
• Apply the Lyapunov's stability theory for the linear and nonlinear systems.
• Know the properties of the solutions of first-order partial differential equations.
• Apply numerical methods to approximate solutions to differential equations.
• Understand the concept of eigenvalues and eigenvectors and use them to solve systems of linear differential equations.

Grading

Course grading range

Grade	Range	Description of performance
A. Excellent	90-100	-
B. Good	75-89	-
C. Satisfactory	60-74	-
D. Fail	0-59	-

Course activities and grading breakdown

Activity Type	Percentage of the overall course grade
Midterm	20
Interim Assessment	20 pts (2 tests by 10 pts)
Final exam	30
Computational assignment	30
Attendance and In-class participation	7

Resources, literature and reference materials

Open access resources

• Elementary Differential Equations by William F. Trench. Brooks/Cole Thomson Learning, 2001 link
• Stephen L. Campbell and Richard Haberman, Introduction to differential equations with dynamical systems
• J.L.Brenner, Problems in DifferentialEquations(adapted from ”Problems in differential equations” by A.F.Filippov)
• S.G.Glebov, O.M.Kiselev, N.Tarkhanov. Nonlinear equations with small parameter. Volume I:Oscillations and resonances