Difference between revisions of "BSc: Differential Equations"

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= Differential Equations =
  
= Differential Equations =
  +
* '''Course name''': Differential Equations
  +
* '''Code discipline''': XYZ
  +
* '''Subject area''': Math
   
  +
== Short Description ==
−
* <span>'''Course name:'''</span> Differential Equations
  
  +
This course covers the following concepts: Ordinary differential equations; Basic numerical methods.
−
* <span>'''Course number:'''</span> XYZ
  
−
* <span>'''Subject area:'''</span> Math
  
   
−
== Course characteristics ==
 +
== Prerequisites ==
   
−
=== Key concepts of the class ===
 +
=== Prerequisite subjects ===
   
−
* Ordinary differential equations
  
−
* Basic numerical methods
  
   
  +
=== Prerequisite topics ===
−
=== What is the purpose of this course? ===
  
  +
  +
  +
== Course Topics ==
  +
{| class="wikitable"
  +
|+ 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.
  
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.
   
−
=== Course Objectives Based on Bloom’s Taxonomy ===
 +
=== ILOs defined at three levels ===
−  
−
=== - What should a student remember at the end of the course? ===
  
   
  +
==== 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,
  
* recognize the type of the equation,
 
* identify the method of analytical solution,
  
* identify the method of analytical solution,
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* match the concrete numerical approach with the necessary level of accuracy.
  
* 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? ===
 +
==== 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,
  
* understand application value of ordinary differential equations,
 
* explain situation when the analytical solution of an equation cannot be found,
  
* explain situation when the analytical solution of an equation cannot be found,
Line 34: Line 81:
 
* restate the given ordinary equation with the Laplace Transform.
  
* restate the given ordinary equation with the Laplace Transform.
   
−
=== - What should a student be able to apply at the end of the course? ===
 +
==== 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),
  
* solve the given ordinary differential equation analytically (if possible),
 
* apply the method of the Laplace Transform for the given initial value problem,
  
* 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,
  
* 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.
 +
* implement a certain numerical method in self-developed computer software.
  +
== Grading ==
   
−
=== Course evaluation ===
 +
=== Course grading range ===
  +
{| class="wikitable"
−  
−
{|
 +
|+
−
|+ Course grade breakdown
  
−
!
  
−
!
  
−
!align="center"| '''Proposed points'''
  
 
|-
  
|-
  +
! Grade !! Range !! Description of performance
−
| Labs/seminar classes
  
−
| 20
  
−
|align="center"| 20
  
 
|-
  
|-
  +
| A. Excellent || 136-170 || -
−
| Interim performance assessment
  
−
| 30
  
−
|align="center"| 70
  
 
|-
  
|-
  +
| B. Good || 102-135 || -
−
| Exams
  
−
| 50
 +
|-
  +
| C. Satisfactory || 68-101 || -
−
|align="center"| 80
  
  +
|-
  +
| D. Poor || 0-68 || -
 
|}
  
|}
   
  +
=== Course activities and grading breakdown ===
−
If necessary, please indicate freely your course’s features in terms of students’ performance assessment:
  
  +
{| class="wikitable"
−  
  +
|+
−
==== 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
  
−
!
  
−
!
  
−
!align="center"| '''Proposed range'''
  
 
|-
  
|-
  +
! Activity Type !! Percentage of the overall course grade
−
| A. Excellent
  
−
| 90-100
  
−
|align="center"| 136-170
  
 
|-
  
|-
  +
| Labs/seminar classes || 20
−
| B. Good
  
−
| 75-89
  
−
|align="center"| 102-135
  
 
|-
  
|-
  +
| Interim performance assessment || 70
−
| C. Satisfactory
  
−
| 60-74
  
−
|align="center"| 68-101
  
 
|-
  
|-
−
| D. Poor
 +
| Exams || 80
−
| 0-59
  
−
|align="center"| 0-68
  
 
|}
  
|}
   
  +
=== Recommendations for students on how to succeed in the course ===
−
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.
  
   
−
=== Resources and reference material ===
 +
== Resources, literature and reference materials ==
   
−
==== Textbook: ====
 +
=== Open access resources ===
   
−
*
 
   
−
==== Reference material: ====
 +
=== Closed access resources ===
   
−
*
 
−
*
 
−
*
 
   
  +
=== Software and tools used within the course ===
−
== Course Sections ==
  
  +
  +
= Teaching Methodology: Methods, techniques, & activities =
   
  +
== Activities and Teaching Methods ==
−
The main sections of the course and approximate hour distribution between them is as follows:
  
  +
{| class="wikitable"
−  
  +
|+ Activities within each section
−
{|
  
−
|+ Course Sections
  
−
|align="center"| '''Section'''
  
−
| '''Section Title'''
  
−
|align="center"| '''Lectures'''
  
−
|align="center"| '''Seminars'''
  
−
|align="center"| '''Self-study'''
  
−
|align="center"| '''Knowledge'''
  
 
|-
  
|-
  +
! Learning Activities !! Section 1 !! Section 2 !! Section 3 !! Section 4 !! Section 5
−
|align="center"| '''Number'''
  
−
|
  
−
|align="center"| '''(hours)'''
  
−
|align="center"| '''(labs)'''
  
−
|align="center"|
  
−
|align="center"| '''evaluation'''
  
 
|-
  
|-
  +
| Homework and group projects || 1 || 1 || 1 || 1 || 1
−
|align="center"| 1
  
−
| First-order equations and their applications
  
−
|align="center"| 12
  
−
|align="center"| 6
  
−
|align="center"| 12
  
−
|align="center"| 4
  
 
|-
  
|-
  +
| Midterm evaluation || 1 || 1 || 1 || 1 || 1
−
|align="center"| 2
  
−
| Introduction to numeric methods for algebraic and first-order differential equations
  
−
|align="center"| 8
  
−
|align="center"| 4
  
−
|align="center"| 22
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Testing (written or computer based) || 1 || 1 || 1 || 1 || 1
−
|align="center"| 3
  
−
| Second-order differential equations and their applications
  
−
|align="center"| 8
  
−
|align="center"| 4
  
−
|align="center"| 8
  
−
|align="center"| 2
  
 
|-
  
|-
  +
| Oral polls || 1 || 0 || 0 || 0 || 0
−
|align="center"| 4
  
−
| Laplace transform
  
−
|align="center"| 8
  
−
|align="center"| 4
  
−
|align="center"| 12
  
−
|align="center"| 3
  
 
|-
  
|-
  +
| Discussions || 1 || 1 || 1 || 1 || 1
−
|align="center"| 5
  
−
| Series approach to linear differential equations
  
−
|align="center"| 8
  
−
|align="center"| 4
  
−
|align="center"| 12
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Development of individual parts of software product code || 0 || 1 || 0 || 0 || 0
−
|align="center"| Final examination
  
−
|
 +
|}
  +
== Formative Assessment and Course Activities ==
−
|align="center"|
  
−
|align="center"|
  
−
|align="center"|
  
−
|align="center"| 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 &amp; 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? ===
 +
=== Ongoing performance assessment ===
   
  +
==== Section 1 ====
−
{|
  
  +
{| class="wikitable"
−
!
  
  +
|+
−
!align="center"| '''Yes/No'''
  
 
|-
  
|-
  +
! Activity Type !! Content !! Is Graded?
−
| Development of individual parts of software product code
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || What is the type of the first order equation? || 1
−
| Homework and group projects
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Is the equation homogeneous or not? || 1
−
| Midterm evaluation
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Which substitution may be used for solving the given equation? || 1
−
| Testing (written or computer based)
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Is the equation linear or not? || 1
−
| Reports
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Which type of the equation have we obtained for the modeled real world problem? || 1
−
| Essays
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Is the equation exact or not? || 1
−
| Oral polls
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Determine the type of the first order equation and solve it with the use of appropriate method. || 0
−
| Discussions
  
  +
|-
−
|align="center"| 1
  
  +
| Question || Find the integrating factor for the given equation. || 0
−
|}
  
  +
|-
−  
  +
| Question || Solve the initial value problem of the first order. || 0
−
=== Typical questions for ongoing performance evaluation within this section ===
  
  +
|-
−  
  +
| 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
−
# What is the type of the first order equation?
  
  +
|}
−
# Is the equation homogeneous or not?
  
  +
==== Section 2 ====
−
# Which substitution may be used for solving the given equation?
  
  +
{| class="wikitable"
−
# 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?
  
  +
! Activity Type !! Content !! Is Graded?
−  
  +
|-
−
=== Typical questions for seminar classes (labs) within this section ===
  
  +
| Question || What is the difference between the methods of sections and tangent line approximations? || 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.
 +
| Question || What is the approximation error for the given method? || 1
  +
|-
−
# Solve the initial value problem of the first order.
  
  +
| Question || How to improve the accuracy of Euler method? || 1
−
# Construct a mathematical model of the presented real world problem in terms of differential equations and answer for the specific question about it.
  
  +
|-
−  
  +
| Question || How to obtain a general formula of the Runge-Kutta methods? || 1
−
=== Test questions for final assessment in this section ===
  
  +
|-
−  
  +
| 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 ====
  +
{| class="wikitable"
  +
|+
  +
|-
  +
! 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 ====
  +
{| class="wikitable"
  +
|+
  +
|-
  +
! 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 ====
  +
{| class="wikitable"
  +
|+
  +
|-
  +
! 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.
  
# Linear first order equation. Integrating factor.
−
# Bernoulli &amp; Riccati equations.
 +
# Bernoulli & Riccati equations.
 
# Homogeneous nonlinear equations equations.
  
# Homogeneous nonlinear equations equations.
 
# Exact equations. Substitutions.
  
# Exact equations. Substitutions.
  +
'''Section 2'''
−  
−
=== 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? ===
  
−  
−
<div class="tabular">
  
−  
−
<span>|a|c|</span> &amp; '''Yes/No'''<br />
  
−
Development of individual parts of software product code &amp; 1<br />
  
−
Homework and group projects &amp; 1<br />
  
−
Midterm evaluation &amp; 1<br />
  
−
Testing (written or computer based) &amp; 0<br />
  
−
Reports &amp; 1<br />
  
−
Essays &amp; 0<br />
  
−
Oral polls &amp; 1<br />
  
−
Discussions &amp; 1<br />
  
−  
−  
−  
−
</div>
  
−
=== Typical questions for ongoing performance evaluation within this section ===
  
−  
−
# 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?
  
−  
−
=== Typical questions for seminar classes (labs) within this section ===
  
−  
−
# 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.
  
−  
−
=== Test questions for final assessment in this section ===
  
−  
 
# Newton’s approximation method.
  
# Newton’s approximation method.
 
# Euler approximation method.
  
# Euler approximation method.
 
# Improved Euler method.
  
# Improved Euler method.
 
# Runge-Kutta methods.
  
# Runge-Kutta methods.
  +
'''Section 3'''
−  
−
=== 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? ===
  
−  
−
<div class="tabular">
  
−  
−
<span>|a|c|</span> &amp; '''Yes/No'''<br />
  
−
Development of individual parts of software product code &amp; 0<br />
  
−
Homework and group projects &amp; 1<br />
  
−
Midterm evaluation &amp; 1<br />
  
−
Testing (written or computer based) &amp; 0<br />
  
−
Reports &amp; 0<br />
  
−
Essays &amp; 0<br />
  
−
Oral polls &amp; 1<br />
  
−
Discussions &amp; 1<br />
  
−  
−  
−  
−
</div>
  
−
=== Typical questions for ongoing performance evaluation within this section ===
  
−  
−
# 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?
  
−  
−
=== Typical questions for seminar classes (labs) within this section ===
  
−  
−
# 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.
  
−  
−
=== Test questions for final assessment in this section ===
  
−  
 
# Homogeneous linear second order equations.
  
# Homogeneous linear second order equations.
 
# Constant coefficient equations. A method of undetermined coefficients.
  
# Constant coefficient equations. A method of undetermined coefficients.
 
# Constant coefficient equations. A method of variation of parameters.
  
# Constant coefficient equations. A method of variation of parameters.
 
# Non-homogeneous linear second order equations. Reduction of order.
  
# Non-homogeneous linear second order equations. Reduction of order.
  +
'''Section 4'''
−  
−
=== 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? ===
  
−  
−
<div class="tabular">
  
−  
−
<span>|a|c|</span> &amp; '''Yes/No'''<br />
  
−
Development of individual parts of software product code &amp; 0<br />
  
−
Homework and group projects &amp; 1<br />
  
−
Midterm evaluation &amp; 0<br />
  
−
Testing (written or computer based) &amp; 1<br />
  
−
Reports &amp; 0<br />
  
−
Essays &amp; 0<br />
  
−
Oral polls &amp; 1<br />
  
−
Discussions &amp; 1<br />
  
−  
−  
−  
−
</div>
  
−
=== Typical questions for ongoing performance evaluation within this section ===
  
−  
−
# What is an improper integral?
  
−
# How to compose the Laplace transform for a certain function?
  
−
# What is a radius of convergence of the Laplace transform?
  
−
# How to determine the inverse Laplace transform for a given expression?
  
−
# How to apply the method of Laplace transform for solving ordinary differential equations?
  
−  
−
=== Typical questions for seminar classes (labs) within this section ===
  
−  
−
# Find the Laplace transform for a given function. Analyze its radius of convergence.
  
−
# Find the inverse Laplace transform for a given expression.
  
−
# Solve the first order differential equation with the use of a Laplace transform.
  
−
# Solve the second order differential equation with the use of a Laplace transform.
  
−  
−
=== Test questions for final assessment in this section ===
  
−  
 
# Laplace transform, its radius of convergence and properties.
  
# Laplace transform, its radius of convergence and properties.
 
# Inverse Laplace transform. The method of rational functions.
  
# Inverse Laplace transform. The method of rational functions.
 
# Application of Laplace transform to solving differential equations.
  
# 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
   
−
=== Section 5 ===
 +
=== The retake exam ===
  +
'''Section 1'''
   
−
==== Section title: ====
 +
'''Section 2'''
   
  +
'''Section 3'''
−
Series approach to linear differential equations
  
   
  +
'''Section 4'''
−
=== Topics covered in this section: ===
  
   
  +
'''Section 5'''
−
* 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? ===
  
−  
−
<div class="tabular">
  
−  
−
<span>|a|c|</span> &amp; '''Yes/No'''<br />
  
−
Development of individual parts of software product code &amp; 0<br />
  
−
Homework and group projects &amp; 1<br />
  
−
Midterm evaluation &amp; 0<br />
  
−
Testing (written or computer based) &amp; 1<br />
  
−
Reports &amp; 0<br />
  
−
Essays &amp; 0<br />
  
−
Oral polls &amp; 1<br />
  
−
Discussions &amp; 1<br />
  
−  
−  
−  
−
</div>
  
−
=== Typical questions for ongoing performance evaluation within this section ===
  
−  
−
# What are the power and functional series?
  
−
# How to find the radius of convergence of a series?
  
−
# What is a Taylor series?
  
−
# How to differentiate a functional series?
  
−  
−
=== Typical questions for seminar classes (labs) within this section ===
  
−  
−
# Find the radius of convergence of a given series.
  
−
# Compose the Taylor series for a given function.
  
−
# Solve the first order differential equation with the use of Series approach.
  
−
# Solve the second order differential equation with the use of Series approach.
  
−  
−
=== Test questions for final assessment in this section ===
  
−  
−
# Taylor and Maclaurin series as functional series. Radius of convergence.
  
−
# Uniqueness of power series. Its differentiation.
  
−
# Application of power series to solving differential equations
  

Latest revision as of 13:19, 13 July 2022

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
  1. The simplest type of differential equation
  2. Separable equation
  3. Initial value problem
  4. Homogeneous nonlinear equations, substitutions
  5. Linear ordinary equations, Bernoulli & Riccati equations
  6. Examples of applications to modeling the real world problems
  7. Exact differential equations, integrating factor
Introduction to numeric methods for algebraic and first-order differential equations
  1. Method of sections (Newton method)
  2. Method of tangent lines approximation (Euler method)
  3. Improved Euler method
  4. Runge-Kutta methods
Second-order differential equations and their applications
  1. Homogeneous linear equations.
  2. Constant coefficient homogeneous equations.
  3. Constant coefficient non-homogeneous equations.
  4. A method of undetermined coefficients.
  5. A method of variation of parameters.
  6. A method of the reduction of order.
Laplace transform
  1. Improper integrals. Convergence / Divergence.
  2. Laplace transform of a function
  3. Existence of the Laplace transform.
  4. Inverse Laplace transform.
  5. Application of the Laplace transform to solving differential equations.
Series approach to linear differential equations
  1. Functional series.
  2. Taylor and Maclaurin series.
  3. Differentiation of power series.
  4. 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

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

Section 2

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

Section 3

  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

  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

  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

The retake exam

Section 1

Section 2

Section 3

Section 4

Section 5

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