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.

What is the main purpose of this course?

The course is designed to provide Software Engineers and Computer Scientists by knowledge of basic (core) concepts, definitions, theoretical results and techniques of ordinary differential equations theory, basics of power series and numerical methods, applications of the all above in sciences. All definitions and theorem statements (that will be given in lectures and that are needed to explain the keywords listed above) will be formal, but just few of these theorems will be proven formally. Instead (in the tutorial and practice classes) we will try these definitions and theorems on work with routine exercises and applied problems.

ILOs defined at three levels

We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

• understand application value of ordinary differential equations,
• explain situation when the analytical solution of an equation cannot be found,
• give the examples of functional series for certain simple functions,
• describe the common goal of the numeric methods,
• restate the given ordinary equation with the Laplace Transform.

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

• recognize the type of the equation,
• identify the method of analytical solution,
• define an initial value problem,
• list alternative approaches to solving ordinary differential equations,
• match the concrete numerical approach with the necessary level of accuracy.

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

By the end of the course, the students should be able to ...

• solve the given ordinary differential equation analytically (if possible),
• apply the method of the Laplace Transform for the given initial value problem,
• predict the number of terms in series solution of the equation depending on the given accuracy,
• implement a certain numerical method in self-developed computer software.

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

Recommendations for students on how to succeed in the course

• Participation is important. Attending lectures is the key to success in this course.
• Review lecture materials before classes to do well.
• Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.

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

Activities and Teaching Methods

Teaching and Learning Methods within each section

Teaching Techniques	Section 1	Section 2	Section 3
Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution)	1	1	1
Project-based learning (students work on a project)	0	1	0
Modular learning (facilitated self-study)	0	0	0
Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level)	1	1	1
Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them)	0	0	0
Business game (learn by playing a game that incorporates the principles of the material covered within the course)	0	0	0
Inquiry-based learning	0	0	0
Just-in-time teaching	0	0	0
Process oriented guided inquiry learning (POGIL)	0	0	0
Studio-based learning	0	0	0
Universal design for learning	0	0	0
Task-based learning	0	0	0

Activities within each section

Learning Activities	Section 1	Section 2	Section 3
Lectures	1	1	1
Interactive Lectures	1	1	1
Lab exercises	1	1	1
Experiments	0	0	0
Modeling	0	0	0
Cases studies	0	0	0
Development of individual parts of software product code	0	0	0
Individual Projects	0	1	0
Group projects	0	0	0
Flipped classroom	0	0	0
Quizzes (written or computer based)	1	1	1
Peer Review	0	0	0
Discussions	1	1	1
Presentations by students	0	0	0
Written reports	0	0	0
Simulations and role-plays	0	0	0
Essays	0	0	0
Oral Reports	0	0	0

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

1. What is the type of the first order equation?
2. Is the equation homogeneous or not?
3. Which substitution may be used for solving the given equation?
4. Is the equation linear or not?
5. Which type of the equation have we obtained for the modeled real world problem?
6. Is the equation exact or not?

Section 2

1. What is the difference between the methods of sections and tangent line approximations?
2. What is the approximation error for the given method?
3. How to improve the accuracy of Euler method?
4. How to obtain a general formula of the Runge-Kutta methods?

Section 3

1. What is the type of the second order equation?
2. Is the equation homogeneous or not?
3. What is a characteristic equation of differential equation?
4. In which form a general solution may be found?
5. What is the form of the particular solution of non-homogeneous equation?
6. How to compose the Laplace transform for a certain function?
7. How to apply the method of Laplace transform for solving ordinary differential equations?
8. How to differentiate a functional series?

Final assessment

Section 1

1. Determine the type of the first order equation and solve it with the use of appropriate method.
2. Find the integrating factor for the given equation.
3. Solve the initial value problem of the first order.
4. Construct a mathematical model of the presented real world problem in terms of differential equations and answer for the specific question about it.

Section 2

1. For the given initial value problem with the ODE of the first order implement in your favorite programming Euler, improved Euler and general Runge-Kutta methods of solving.
2. Using the developed software construct corresponding approximation of the solution of a given initial value problem (provide the possibility of changing of the initial conditions, implement the exact solution to be able to compare the obtained results).
3. Investigate the convergence of the numerical methods on different grid sizes.
4. Compare approximation errors of these methods plotting the corresponding chart for the dependency of approximation error on a grid size.

Section 3

1. Compose a characteristic equation and find its roots.
2. Find the general of second order equation.
3. Determine the form of a particular solution of the equation and reduce the order.
4. Solve a homogeneous constant coefficient equation.
5. Solve a non-homogeneous constant coefficient equation.
6. Find the Laplace transform for a given function. Analyze its radius of convergence.
7. Find the inverse Laplace transform for a given expression.
8. Solve the second order differential equation with the use of a Laplace transform.
9. Solve the second order differential equation with the use of Series approach.

The retake exam

Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.