Difference between revisions of "BSc: Differential Equations.f22"

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==== Section 1 ====
  
==== Section 1 ====
  +
# What is the type of the first order equation?
−
# A plane curve is given by <math>x(t)=-\frac{t^2+4t+8}{t+2}</math>, <math display="inline">y(t)=\frac{t^2+9t+22}{t+6}</math>. Find
  
−
#: the asymptotes of this curve;
 +
# Is the equation homogeneous or not?
  +
# Which substitution may be used for solving the given equation?
−
#: the derivative <math display="inline">y'_x</math>.
  
  +
# Is the equation linear or not?
−
# Apply Leibniz formula Find <math display="inline">y^{(n)}(x)</math> if <math display="inline">y(x)=\left(x^2-2\right)\cos2x\sin3x</math>.
  
  +
# Which type of the equation have we obtained for the modeled real world problem?
−
#: Draw graphs of functions
  
  +
# Is the equation exact or not?
−
#: Find asymptotes
  
−
# Find the derivatives of the following functions:
  
−
#* <math display="inline">f(x)=\log_{|\sin x|}\sqrt[6]{x^2+6}</math>;
  
−
#* <math display="inline">y(x)</math> that is given implicitly by <math display="inline">x^3+5xy+y^3=0</math>.
  
   
 
==== Section 2 ====
  
==== Section 2 ====
  +
# What is the difference between the methods of sections and tangent line approximations?
−
# Find the following integrals:
  
  +
# What is the approximation error for the given method?
−
#*<math display="inline">\int\frac{\sqrt{4+x^2}+2\sqrt{4-x^2}}{\sqrt{16-x^4}}\,dx</math>;
  
  +
# How to improve the accuracy of Euler method?
−
#*<math display="inline">\int2^{2x}e^x\,dx</math>;
  
  +
# How to obtain a general formula of the Runge-Kutta methods?
−
#*<math display="inline">\int\frac{dx}{3x^2-x^4}</math>.
  
−
# Find the indefinite integral <math display="inline">\displaystyle\int x\ln\left(x+\sqrt{x^2-1}\right)\,dx</math>.
  
−
# Find the length of a curve given by <math display="inline">y=\ln\sin x</math>, <math display="inline">\frac{\pi}4\leqslant x\leqslant\frac{\pi}2</math>.
  
   
 
==== Section 3 ====
  
==== Section 3 ====
  +
# What is the type of the second order equation?
−
# Find limits of the following sequences or prove that they do not exist:
  
  +
# Is the equation homogeneous or not?
−
#* <math>a_n=n-\sqrt{n^2-70n+1400}</math>;
  
  +
# What is a characteristic equation of differential equation?
−
#* <math display="inline">d_n=\left(\frac{2n-4}{2n+1}\right)^{n}</math>;
  
  +
# In which form a general solution may be found?
−
#* <math display="inline">x_n=\frac{\left(2n^2+1\right)^6(n-1)^2}{\left(n^7+1000n^6-3\right)^2}</math>.
  
  +
# 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 ===
  
=== Final assessment ===
−
'''Section 1'''
 +
==== Section 1 ====
  +
# Determine the type of the first order equation and solve it with the use of appropriate method.
−
# Apply the appropriate differentiation technique to a given problem.
  
−
# Find a derivative of a function
 +
# Find the integrating factor for the given equation.
  +
# Solve the initial value problem of the first order.
−
# Apply Leibniz formula
  
  +
# Construct a mathematical model of the presented real world problem in terms of differential equations and answer for the specific question about it.
−
# Draw graphs of functions
  
  ⚫
==== Section 2 ====
−
# Find asymptotes of a parametric function
  
  +
# 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).
⚫
'''Section 2'''
  
  +
# Investigate the convergence of the numerical methods on different grid sizes.
−
# Apply the appropriate integration technique to the given problem
  
  +
# Compare approximation errors of these methods plotting the corresponding chart for the dependency of approximation error on a grid size.
−
# Find the value of the devinite integral
  
  ⚫
==== Section 3 ====
−
# Calculate the area of the domain or the length of the curve
  
  +
# Compose a characteristic equation and find its roots.
−  
  +
# Find the general of second order equation.
⚫
'''Section 3'''
  
  +
# Determine the form of a particular solution of the equation and reduce the order.
−
# Find a limit of a sequence
  
  +
# Solve a homogeneous constant coefficient equation.
−
# Find a limit of a function
  
  +
# 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 ===
  
=== The retake exam ===

Latest revision as of 11:16, 28 June 2022

Differential Equations

  • Course name: Differential Equations
  • Code discipline: CSE205
  • Subject area: Math

Short Description

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 Topics

Course Sections and Topics
Section Topics within the section
First-order equations and their applications
  1. Separable equation
  2. Initial value problem
  3. Homogeneous nonlinear equations
  4. Substitutions
  5. Linear ordinary equations
  6. Bernoulli & Riccati equations
  7. Exact differential equations, integrating factor
  8. Examples of applications to modeling the real world problems
Introduction to Numerical Methods
  1. Method of sections (Newton method)
  2. Method of tangent lines (Euler method)
  3. Improved Euler method
  4. Runge-Kutta methods
Higher-order equations and systems
  1. Homogeneous linear equations
  2. Constant coefficient equations
  3. A method of undetermined coefficients
  4. A method of variation of parameters
  5. A method of the reduction of order
  6. Laplace transform. Inverse Laplace transform.
  7. Application of the Laplace transform to solving differential equations.
  8. Series solution of differential equations.
  9. Homogeneous linear systems
  10. Non-homogeneous systems
  11. Matrices, eigenvalues and matrix form of the systems of ODE

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

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
Final exam 30
Computational assignment 25
In-class participation 5

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

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.

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