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
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
- What is the type of the first order equation?
- Is the equation homogeneous or not?
- Which substitution may be used for solving the given equation?
- Is the equation linear or not?
- Which type of the equation have we obtained for the modeled real world problem?
- Is the equation exact or not?
Section 2
- What is the difference between the methods of sections and tangent line approximations?
- What is the approximation error for the given method?
- How to improve the accuracy of Euler method?
- How to obtain a general formula of the Runge-Kutta methods?
Section 3
- What is the type of the second order equation?
- Is the equation homogeneous or not?
- What is a characteristic equation of differential equation?
- In which form a general solution may be found?
- What is the form of the particular solution of non-homogeneous equation?
- How to compose the Laplace transform for a certain function?
- How to apply the method of Laplace transform for solving ordinary differential equations?
- How to differentiate a functional series?
Final assessment
Section 1
- Determine the type of the first order equation and solve it with the use of appropriate method.
- Find the integrating factor for the given equation.
- Solve the initial value problem of the first order.
- 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
- 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.
- 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).
- Investigate the convergence of the numerical methods on different grid sizes.
- Compare approximation errors of these methods plotting the corresponding chart for the dependency of approximation error on a grid size.
Section 3
- Compose a characteristic equation and find its roots.
- Find the general of second order equation.
- Determine the form of a particular solution of the equation and reduce the order.
- Solve a homogeneous constant coefficient equation.
- Solve a non-homogeneous constant coefficient equation.
- Find the Laplace transform for a given function. Analyze its radius of convergence.
- Find the inverse Laplace transform for a given expression.
- Solve the second order differential equation with the use of a Laplace transform.
- 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.