BSc: Differential Equations

Differential Equations

• Course name: Differential Equations
• Course number: XYZ
• Subject area: Math

Course characteristics

Key concepts of the class

• Ordinary differential equations
• Basic numerical methods

What is the 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.

- What should a student remember at the end of the course?

• 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.

- What should a student be able to understand at the end of the course?

• 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.

- What should a student be able to apply at the end of the course?

• 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.

Course evaluation

Course grade breakdown

		Proposed points
Labs/seminar classes	20	20
Interim performance assessment	30	70
Exams	50	80

If necessary, please indicate freely your course’s features in terms of students’ performance assessment:

Labs/seminar classes:

• In-class participation 1 point for each individual contribution in a class but not more than 1 point a week (i.e. 14 points in total for 14 study weeks),
• overall course contribution (to accumulate extra-class activities valuable to the course progress, e.g. a short presentation, book review, very active in-class participation, etc.) up to 6 points.

Interim performance assessment:

• in-class tests up to 10 points for each test (i.e. up to 40 points in total for 2 theory and 2 practice tests),
• computational practicum assignment up to 10 points for each task (i.e. up to 30 points for 3 tasks).

Exams:

• mid-term exam up to 40 points,
• final examination up to 40 points.

Overall score:

170 points (100%).

Grades range

Course grading range

		Proposed range
A. Excellent	90-100	136-170
B. Good	75-89	102-135
C. Satisfactory	60-74	68-101
D. Poor	0-59	0-68

If necessary, please indicate freely your course’s grading features:

• A: at least 80% of the overall score;
• B: at least 60% of the overall score;
• C: at least 40% of the overall score;
• D: less than 40% of the overall score.

Course Sections

The main sections of the course and approximate hour distribution between them is as follows:

Course Sections

Section	Section Title	Lectures	Seminars	Self-study	Knowledge
Number		(hours)	(labs)		evaluation
1	First-order equations and their applications	12	6	12	4
2	Introduction to numeric methods for algebraic and first-order differential equations	8	4	22	1
3	Second-order differential equations and their applications	8	4	8	2
4	Laplace transform	8	4	12	3
5	Series approach to linear differential equations	8	4	12	0
Final examination					2

Section 1

Section title:

First-order equations and their applications

Topics covered in this section:

• 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

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 0
Homework and group projects & 1
Midterm evaluation & 1
Testing (written or computer based) & 1
Reports & 0
Essays & 0
Oral polls & 1
Discussions & 1

Typical questions for ongoing performance evaluation within this section

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?

Typical questions for seminar classes (labs) within this section

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.

Test questions for final assessment in this section

1. Linear first order equation. Integrating factor.
2. Bernoulli & Riccati equations.
3. Homogeneous nonlinear equations equations.
4. Exact equations. Substitutions.

Section 2

Section title:

Introduction to numeric methods for algebraic and first-order differential equations

Topics covered in this section:

• Method of sections (Newton method)
• Method of tangent lines approximation (Euler method)
• Improved Euler method
• Runge-Kutta methods

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 1
Homework and group projects & 1
Midterm evaluation & 1
Testing (written or computer based) & 0
Reports & 1
Essays & 0
Oral polls & 1
Discussions & 1

Typical questions for ongoing performance evaluation within this section

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?

Typical questions for seminar classes (labs) within this section

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.

Test questions for final assessment in this section

1. Newton’s approximation method.
2. Euler approximation method.
3. Improved Euler method.
4. Runge-Kutta methods.

Section 3

Section title:

Second-order differential equations and their applications

Topics covered in this section:

• 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.

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 0
Homework and group projects & 1
Midterm evaluation & 1
Testing (written or computer based) & 0
Reports & 0
Essays & 0
Oral polls & 1
Discussions & 1

Typical questions for ongoing performance evaluation within this section

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?

Typical questions for seminar classes (labs) within this section

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.

Test questions for final assessment in this section

1. Homogeneous linear second order equations.
2. Constant coefficient equations. A method of undetermined coefficients.
3. Constant coefficient equations. A method of variation of parameters.
4. Non-homogeneous linear second order equations. Reduction of order.

Section 4

Section title:

Laplace transform

Topics covered in this section:

• 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.

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 0
Homework and group projects & 1
Midterm evaluation & 0
Testing (written or computer based) & 1
Reports & 0
Essays & 0
Oral polls & 1
Discussions & 1

Typical questions for ongoing performance evaluation within this section

1. What is an improper integral?
2. How to compose the Laplace transform for a certain function?
3. What is a radius of convergence of the Laplace transform?
4. How to determine the inverse Laplace transform for a given expression?
5. How to apply the method of Laplace transform for solving ordinary differential equations?

Typical questions for seminar classes (labs) within this section

1. Find the Laplace transform for a given function. Analyze its radius of convergence.
2. Find the inverse Laplace transform for a given expression.
3. Solve the first order differential equation with the use of a Laplace transform.
4. Solve the second order differential equation with the use of a Laplace transform.

Test questions for final assessment in this section

1. Laplace transform, its radius of convergence and properties.
2. Inverse Laplace transform. The method of rational functions.
3. Application of Laplace transform to solving differential equations.

Section 5

Section title:

Series approach to linear differential equations

Topics covered in this section:

• Functional series.
• Taylor and Maclaurin series.
• Differentiation of power series.
• Series solution of differential equations.

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 0
Homework and group projects & 1
Midterm evaluation & 0
Testing (written or computer based) & 1
Reports & 0
Essays & 0
Oral polls & 1
Discussions & 1

Typical questions for ongoing performance evaluation within this section

1. What are the power and functional series?
2. How to find the radius of convergence of a series?
3. What is a Taylor series?
4. How to differentiate a functional series?

Typical questions for seminar classes (labs) within this section

1. Find the radius of convergence of a given series.
2. Compose the Taylor series for a given function.
3. Solve the first order differential equation with the use of Series approach.
4. Solve the second order differential equation with the use of Series approach.

Test questions for final assessment in this section

1. Taylor and Maclaurin series as functional series. Radius of convergence.
2. Uniqueness of power series. Its differentiation.
3. Application of power series to solving differential equations