Differential Equations
- Course name: Differential Equations
- Code discipline: XYZ
- Subject area: Math
Short Description
This course covers the following concepts: Ordinary differential equations; Basic numerical methods.
Prerequisites
Prerequisite subjects
Prerequisite topics
Course Topics
Course Sections and Topics
| Section |
Topics within the section
|
| First-order equations and their applications |
- The simplest type of differential equation
- Separable equation
- Initial value problem
- Homogeneous nonlinear equations, substitutions
- Linear ordinary equations, Bernoulli & Riccati equations
- Examples of applications to modeling the real world problems
- Exact differential equations, integrating factor
|
| Introduction to numeric methods for algebraic and first-order differential equations |
- Method of sections (Newton method)
- Method of tangent lines approximation (Euler method)
- Improved Euler method
- Runge-Kutta methods
|
| Second-order differential equations and their applications |
- Homogeneous linear equations.
- Constant coefficient homogeneous equations.
- Constant coefficient non-homogeneous equations.
- A method of undetermined coefficients.
- A method of variation of parameters.
- A method of the reduction of order.
|
| Laplace transform |
- Improper integrals. Convergence / Divergence.
- Laplace transform of a function
- Existence of the Laplace transform.
- Inverse Laplace transform.
- Application of the Laplace transform to solving differential equations.
|
| Series approach to linear differential equations |
- Functional series.
- Taylor and Maclaurin series.
- Differentiation of power series.
- Series solution of differential equations.
|
Intended Learning Outcomes (ILOs)
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
Level 1: What concepts should a student know/remember/explain?
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 2: What basic practical skills should a student be able to perform?
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 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 |
136-170 |
-
|
| B. Good |
102-135 |
-
|
| C. Satisfactory |
68-101 |
-
|
| D. Poor |
0-68 |
-
|
Course activities and grading breakdown
| Activity Type |
Percentage of the overall course grade
|
| Labs/seminar classes |
20
|
| Interim performance assessment |
70
|
| Exams |
80
|
Recommendations for students on how to succeed in the course
Resources, literature and reference materials
Open access resources
Closed access resources
Software and tools used within the course
Teaching Methodology: Methods, techniques, & activities
Activities and Teaching Methods
Activities within each section
| Learning Activities |
Section 1 |
Section 2 |
Section 3 |
Section 4 |
Section 5
|
| Homework and group projects |
1 |
1 |
1 |
1 |
1
|
| Midterm evaluation |
1 |
1 |
1 |
1 |
1
|
| Testing (written or computer based) |
1 |
1 |
1 |
1 |
1
|
| Oral polls |
1 |
0 |
0 |
0 |
0
|
| Discussions |
1 |
1 |
1 |
1 |
1
|
| Development of individual parts of software product code |
0 |
1 |
0 |
0 |
0
|
Formative Assessment and Course Activities
Ongoing performance assessment
Section 1
| Activity Type |
Content |
Is Graded?
|
| Question |
What is the type of the first order equation? |
1
|
| Question |
Is the equation homogeneous or not? |
1
|
| Question |
Which substitution may be used for solving the given equation? |
1
|
| Question |
Is the equation linear or not? |
1
|
| Question |
Which type of the equation have we obtained for the modeled real world problem? |
1
|
| Question |
Is the equation exact or not? |
1
|
| Question |
Determine the type of the first order equation and solve it with the use of appropriate method. |
0
|
| Question |
Find the integrating factor for the given equation. |
0
|
| Question |
Solve the initial value problem of the first order. |
0
|
| Question |
Construct a mathematical model of the presented real world problem in terms of differential equations and answer for the specific question about it. |
0
|
Section 2
| Activity Type |
Content |
Is Graded?
|
| Question |
What is the difference between the methods of sections and tangent line approximations? |
1
|
| Question |
What is the approximation error for the given method? |
1
|
| Question |
How to improve the accuracy of Euler method? |
1
|
| Question |
How to obtain a general formula of the Runge-Kutta methods? |
1
|
| Question |
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. |
0
|
| Question |
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). |
0
|
| Question |
Investigate the convergence of the numerical methods on different grid sizes. |
0
|
| Question |
Compare approximation errors of these methods plotting the corresponding chart for the dependency of approximation error on a grid size. |
0
|
Section 3
| Activity Type |
Content |
Is Graded?
|
| Question |
What is the type of the second order equation? |
1
|
| Question |
Is the equation homogeneous or not? |
1
|
| Question |
What is a characteristic equation of differential equation? |
1
|
| Question |
In which form a general solution may be found? |
1
|
| Question |
What is the form of the particular solution of non-homogeneous equation? |
1
|
| Question |
Compose a characteristic equation and find its roots. |
0
|
| Question |
Find the general of second order equation. |
0
|
| Question |
Determine the form of a particular solution of the equation and reduce the order. |
0
|
| Question |
Solve a homogeneous constant coefficient equation. |
0
|
| Question |
Solve a non-homogeneous constant coefficient equation. |
0
|
Section 4
| Activity Type |
Content |
Is Graded?
|
| Question |
What is an improper integral? |
1
|
| Question |
How to compose the Laplace transform for a certain function? |
1
|
| Question |
What is a radius of convergence of the Laplace transform? |
1
|
| Question |
How to determine the inverse Laplace transform for a given expression? |
1
|
| Question |
How to apply the method of Laplace transform for solving ordinary differential equations? |
1
|
| Question |
Find the Laplace transform for a given function. Analyze its radius of convergence. |
0
|
| Question |
Find the inverse Laplace transform for a given expression. |
0
|
| Question |
Solve the first order differential equation with the use of a Laplace transform. |
0
|
| Question |
Solve the second order differential equation with the use of a Laplace transform. |
0
|
Section 5
| Activity Type |
Content |
Is Graded?
|
| Question |
What are the power and functional series? |
1
|
| Question |
How to find the radius of convergence of a series? |
1
|
| Question |
What is a Taylor series? |
1
|
| Question |
How to differentiate a functional series? |
1
|
| Question |
Find the radius of convergence of a given series. |
0
|
| Question |
Compose the Taylor series for a given function. |
0
|
| Question |
Solve the first order differential equation with the use of Series approach. |
0
|
| Question |
Solve the second order differential equation with the use of Series approach. |
0
|
Final assessment
Section 1
- Linear first order equation. Integrating factor.
- Bernoulli & Riccati equations.
- Homogeneous nonlinear equations equations.
- Exact equations. Substitutions.
Section 2
- Newton’s approximation method.
- Euler approximation method.
- Improved Euler method.
- Runge-Kutta methods.
Section 3
- Homogeneous linear second order equations.
- Constant coefficient equations. A method of undetermined coefficients.
- Constant coefficient equations. A method of variation of parameters.
- Non-homogeneous linear second order equations. Reduction of order.
Section 4
- Laplace transform, its radius of convergence and properties.
- Inverse Laplace transform. The method of rational functions.
- Application of Laplace transform to solving differential equations.
Section 5
- Taylor and Maclaurin series as functional series. Radius of convergence.
- Uniqueness of power series. Its differentiation.
- Application of power series to solving differential equations
The retake exam
Section 1
Section 2
Section 3
Section 4
Section 5