BSc: Differential Equations.f22

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	Separable equation Initial value problem Homogeneous nonlinear equations Substitutions Linear ordinary equations Bernoulli & Riccati equations Exact differential equations, integrating factor Examples of applications to modeling the real world problems
Introduction to Numerical Methods	Method of sections (Newton method) Method of tangent lines (Euler method) Improved Euler method Runge-Kutta methods
Higher-order equations and systems	Homogeneous linear equations Constant coefficient equations A method of undetermined coefficients A method of variation of parameters A method of the reduction of order Laplace transform. Inverse Laplace transform. Application of the Laplace transform to solving differential equations. Series solution of differential equations. Homogeneous linear systems Non-homogeneous systems 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	0	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. A plane curve is given by $x(t)=-{\frac {t^{2}+4t+8}{t+2}}$, ${\textstyle y(t)={\frac {t^{2}+9t+22}{t+6}}}$. Find

   the asymptotes of this curve;

   the derivative ${\textstyle y'_{x}}$.

2. Apply Leibniz formula Find ${\textstyle y^{(n)}(x)}$ if ${\textstyle y(x)=\left(x^{2}-2\right)\cos 2x\sin 3x}$.

   Draw graphs of functions

   Find asymptotes

3. Find the derivatives of the following functions:
   • ${\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}$;
   • ${\textstyle y(x)}$ that is given implicitly by ${\textstyle x^{3}+5xy+y^{3}=0}$.

Section 2

1. Find the following integrals:
   • ${\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}$;
   • ${\textstyle \int 2^{2x}e^{x}\,dx}$;
   • ${\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}$.
2. Find the indefinite integral ${\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}$.
3. Find the length of a curve given by ${\textstyle y=\ln \sin x}$, ${\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}$.

Section 3

1. Find limits of the following sequences or prove that they do not exist:
   • $a_{n}=n-{\sqrt {n^{2}-70n+1400}}$;
   • ${\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}$;
   • ${\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}$.

Final assessment

Section 1

1. Apply the appropriate differentiation technique to a given problem.
2. Find a derivative of a function
3. Apply Leibniz formula
4. Draw graphs of functions
5. Find asymptotes of a parametric function

Section 2

1. Apply the appropriate integration technique to the given problem
2. Find the value of the devinite integral
3. Calculate the area of the domain or the length of the curve

Section 3

1. Find a limit of a sequence
2. Find a limit of a function

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.