Course: Mathematics-III: Ordinary and Partial Differential Equations (Summer 20)

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Teacher Information
Md. Shorif Hossan (MSH) Lecturer (Mathematics) Department Of General Educational Development (GED), FSIT, DIU. Counseling Time: Anytime Teachers Room: 505 (AB-4), PC, DIU. Cell: 01517-814722 E-Mail: shorif.ged0175.c@diu.edu.bd

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Select activity IntroductionToday's fast-moving information techno...

Introduction
Today's fast-moving information technology field requires professionals with math skills that will require them to manipulate mathematical concepts and interpret data. Computer Networking and database technologies require knowledge in Ordinary and partial differential equations, and this course will give the students a good working knowledge of these areas.
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
Mathematics III is a branch of applied mathematics concerning mathematical methods and techniques that are typically used in computer engineering and industry. Along with fields like engineering physics and engineering geology, both of which may belong in the wider category engineering science, engineering mathematics is an interdisciplinary subject motivated by engineers' needs both for practical, theoretical and other considerations out with their specialization and to deal with constraints to be effective in their work. Historically, mathematics III consisted mostly of applied analysis, most notably: differential equations; ordinary differential equation, and partial differential equations. The success of modern numerical computer methods and software has led to the emergence of computational mathematics, computational science, and computational engineering, which occasionally use high-performance computing for the simulation of phenomena and the solution of problems in the sciences and engineering. These are often considered interdisciplinary fields but are also of interest to mathematics III. Engineering mathematics in tertiary education typically consists of mathematical methods and model courses.
Select activity Objective/Goals to the course:A mathematics course...

Objective/Goals to the course:
A mathematics course has more than one purpose. Students should learn a particular set of mathematical facts and how to apply them; more importantly, such a course should teach students how to think logically and mathematically. To achieve these goals, this text stresses mathematical reasoning and the different ways problems are solved.

Ø Students will be able to apply the concepts and methods described in the syllabus

Ø They will be able to solve problems using the differential equation

Ø They will know a number of applications.

Ø The text and class discussion will introduce the concepts, methods, applications, and logical arguments;

Ø Students will practice them and solve problems on daily assignments, and they will be tested on quizzes, midterms, and the final.
Select activity Outcomes of the course:On successful completion of...
Outcomes of the course:
On successful completion of this course students will be able to:
Apply statistical analysis of a variety of experimental and observational studies
Solve statistical problems using computational tools.
Derive mathematical models of physical systems.
Solve differential equations using appropriate methods.
Present mathematical solutions in a concise and informative manner.
develop a logical understanding of the subject.
develop mathematical skills so that students are able to apply mathematical methods & principals in solving problems from Engineering fields.
identify the pattern of the program outcomes on engineering mathematics subjects.
explain the concept of a differential equation and Classify the differential equations with respect to their order and linearity.
solve first-order ordinary differential equations, Convert, separable and homogenous equations to exact differential equations by integrating factors, Solve Bernoulli differential equations.
find solutions to higher-order linear differential equations.
classify partial differential equations and transform into canonical form
solve linear partial differential equations of both first and second order
apply partial derivative equation techniques to predict the behaviour of certain phenomena.
apply specific methodologies, techniques and resources to conduct research and produce innovative results in the area of specialisation.
extract information from partial derivative models in order to interpret reality.
identify real phenomena as models of partial derivative equations.
Select activity Course Contents:Ordinary Differential Equation: F...

Course Contents:
Ordinary Differential Equation: Formation of Differential Equation; First order and a first-degree differential equation, Separation of Variables, Homogenous equation, Equation reducible to homogenous, Exact equation, Linear Equation, Reducible to Linear Equation, Linear Differential Equations with Constant Coefficients, Linear Differential Equation with right-hand side non zero, Variation of parameter, Method of Successive approximation, Reduction of Order, Method of undermined Coefficient, Matrix method, Various types of Application of Differential Equations

Partial Differential Equation: Formation of Partial Differential equation, Linear and Non-Linear first-order equation, Standard forms, Linear Equation of higher order, Partial Differential Equations With Constant Coefficients, Equation of second order with variable coefficients, Wave & Heat equations, Particular solution with boundary and initial conditions.

Assessment plan:

Course Assessment Plan

Assessment Policy:

Assessment	Probable Date	Syllabus	Questions	MarkS
Quiz 01	Will be declared in the class.	Class Lecture	Five Short Question	3*5=15
Quiz 02	Will be declared in the class.
Quiz 03	Will be declared in the class.
Assignment	Based on the class lectures	Will be given in class		(5 Marks)
Presentation	After the Midterm exam	The topic will be announced in the class		(8 Marks)
Midterm Exam	According to the date declared by the University	Lecture 01-13	Five out of six questions	5*5=25
Final Exam	According to the date declared by the University	Lecture 14-24	Five out of six questions	5*8=40
Class Attendance	Marks will be distributed equally according to the number of classes attended (7 Marks)

Grading Policy:

Marks out of 100	Letter Grade	Grade Point
80 - 100	A+	4.00
75 - 79	A	3.75
70 - 74	A-	3.50
65 - 69	B+	3.25
60 - 64	B	3.00
55 - 59	B-	2.75
50 - 54	C+	2.50
45 - 49	C	2.25
40 – 44	D	2.00
00 - 39	F	0.00

ü Besides, ‘W’ and ‘I’ grade may be awarded as per university rule.

Marks distribution:

The final course grade will be awarded based on the marks distribution shown in the table above. Percentages of marks for the different heads are given below

Attendance	07 %
Class Test	15 %
Assignment	05%
Presentation	08%
Midterm	25%
Final Exam	40 %
Total	100%

Semester Class Schedule

SESSION	DURATION	COURSE CONTENT
Session 1	90 minutes	Introductory Class some pre-requisite topics will be discussed
Session 2	90 minutes	Differential Equations: Basic concepts,
Session 3	90 minutes	A solution of the first-order and first degree ODE, Separation of variables.
Session 4	90 minutes	A solution of Homogeneous Equation
Session 5	90 minutes	Reducible to the homogeneous equation
Session 6	90 minutes	A solution of 1st order Linear equation with IF factor
Session 7	90 minutes	Quiz-1 and discussion
Session 8	90 minutes	A solution of Bernoulli’s equations
Session 9	90 minutes	A solution of linear model differential equations
Session 10	90 minutes	Properties of Solutions of general linear equations with constant coefficients
Session 11	90 minutes	Math Solutions of general linear equations with constant coefficients
Session 12	90 minutes	Math Solutions of general linear equations with constant coefficients: right side non zero
Session 13	90 minutes	Quiz 2 and discussion
Mid Term Examination
Session 14	90 minutes	The solution to the initial value problem
Session 15	90 minutes	Application of linear equations of 1st order
Session 16	90 minutes	Application of linear equations of 2nd order
Session 17	90 minutes	Introduction to the PDE
Session 18	90 minutes	Formation to the PDE
Session 19	90 minutes	Quiz 3 and discussion
Session 20	90 minutes	1^st order PDE: Lagrange equation
Session 21	90 minutes	PDE with constant coefficient with right-hand side zero
Session 22	90 minutes	PDE with constant coefficient with right-hand side non zero
Session 23	90 minutes	Wave and heat equation
Session 24	90 minutes	Revision of lectures
Final Examination

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COURSE OUTLINE File
Select activity References:1. A First Course In Differential Equat...

References:
1. A First Course In Differential Equations With Modeling Applications by Dennis G. Zill
2. Differential Equations by Shepley L. Ross
3. Differential Equations by Paul Dawkins
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Some helpful materials File
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Google Classroom:
https://classroom.google.com/u/0/c/MTA4NTg1NDE0MzI2
class code: ojaeqww

Select section Lecture #1

Lecture #1
Introductory class:

We will discuss ourselves
Select section lecture #2

lecture #2
Ordinary Differential Equation
Introduction:
In this class, an Ordinary differential equation will be discussed. In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
Objectives:
After studying this unit you will be able to
know about differential equation
know about the formation of differential equation
use equations to solve a real-life problem
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Formula of differentiation and integration
  
  Formula of differentiation and integration File
- Select activity Introduction to differential equation and formation
  
  Introduction to differential equation and formation File
Select section LECTURE 3

LECTURE 3
Order, degree, and variable separable
Introduction:
In this class, the order and degree of a differential equation will be discussed.
Let a differential equation be expressed as a polynomial equation in the derivatives occurring in it.
The exponent of each of the derivatives should be minimum. The exponent of the variables need not be an integer.
Under such conditions,
i) the order of the differential equation is the order of the highest derivative in it.
ii) the degree of the differential equation is the largest exponent of the highest order derivative
Separable equations introduction. "Separation of variables" allows us to rewrite differential equations so we obtain equality between two integrals we can evaluate. Separable equations are the class of differential equations that can be solved using this method.
Objectives:
order and degree of ordinary differential equation
solve 1st order ODE by using variable separation method
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Separation of variable
  
  Separation of variable File
Select section LECTURE 4-5

LECTURE 4-5
Homogeneous Differential Equation

Introduction:
In this class, a Homogeneous differential equation will be discussed.
A differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term. A homogeneous differential equation is of prime importance in the physical applications of mathematics due to its simple structure and useful solution.
Objectives:
know about homogeneous equation
solving method of homogeneous equations
application of homogeneous equation
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note
  
  Lecture note File
Select section LECUTURE 6-7

LECUTURE 6-7
First-order differential equation and integrating factor
Introduction:
In this class, the first order and integrating factor of differential equations will be discussed.
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where the temperature becomes the integrating factor that makes entropy an exact differential.
Objectives:
to generalize the integrating factor method from linear scalar differential equations to a linear system of differential equations
to solve first-order linear differential equation
to make an exact differential equation
solving the real-life problem by IF method
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity 1st order differential equation and integrating factor
  
  1st order differential equation and integrating factor File
Select section Lecture #8

Lecture #8
Select section LECTURE 9

LECTURE 9
Bernoulli differential equation:
Introduction:
In this class, the Bernoulli differential equation will be discussed.
In mathematics, an ordinary differential equation of the form $y'+P(x)y=Q(x)y^{n}$ is called a Bernoulli differential equation where is any real number other than 0 or 1. It is named after Jacob Bernoulli, who discussed it in 1695. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A famous special case of the Bernoulli equation is the logistic differential equation.
Objectives:
know about first-order non-linear differential equation
transforming non-linear to a linear equation
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Bernoulli’s Equation
  
  Bernoulli’s Equation File
Select section LECTURE 10-12

LECTURE 10-12
Linear differential equation with constant coefficients
Introduction:
In this class, a linear differential equation with constant coefficients will be discussed.
If the constant term is the zero function, then the differential equation is said to be homogeneous, as it is a homogeneous polynomial in the unknown function and its derivatives. ... A differential equation has constant coefficients if only constant functions appear as coefficients in the associated homogeneous equation.

Homogeneous equation with constant coefficients
A homogeneous linear differential equation has constant coefficients if it has the form
$a_{0}y+a_{1}y'+a_{2}y''+\cdots +a_{n}y^{(n)}=0$
where $a_1, \ldots, a_n$ are (real or complex) numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.

Non-homogeneous equation with constant coefficients:
A non-homogeneous equation of order n with constant coefficients may be written
$y^{(n)}(x)+a_{1}y^{(n-1)}(x)+\cdots +a_{n-1}y'(x)+a_{n}y(x)=f(x),$
where $a_1, \ldots, a_n$ are real or complex numbers, f is a given function of x, and y is the unknown function (for sake of simplicity, "(x)" will be omitted in the following). There are several methods for solving such an equation. The best method depends on the nature of the function f that makes the equation non-homogeneous. If f is a linear combination of exponential and sinusoidal functions, then the exponential response formula may be used.

Objectives:
Solve the homogeneous linear differential equations with constant coefficients.
Apply the method to solve the non-homogeneous linear differential equations with constant coefficients.
solve systems of linear differential equations.
determine the type of linear differential equation systems
Use the operator method to solve linear systems with constant coefficients.
Solve the linear systems in normal form.
Solve the homogeneous linear systems with constant coefficients.
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note
  
  Lecture note File
Select section Lecture 13

Lecture 13
Select section LECTURE 14

LECTURE 14
Initial value problem
Introduction:
In this class, the Initial value problem will be discussed.
In the field of differential equations, an initial value problem (also called a Cauchy problem by some authors) is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution. In physics or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential initial value is an equation that is an evolution equation specifying how, given initial conditions, the system will evolve with time.
Objectives:
Recognize an initial value problem
solve initial value problem of the linear differential equation with constant coefficients
application of initial value problem
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note
  
  Lecture note File
Select section Midterm Examination

Midterm Examination
Midterm Examination
Select section Lecture 15-16

Lecture 15-16
Application of ODE
Introduction:
In this class, the application of ODE will be discussed.
Objectives:
to know about real-life application of ODE: Exponential Growth - Population, Exponential Decay - Radioactive Material, Falling Object, Newton's Law of Cooling, RL circuit
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note
  
  Lecture note File
Select section LECTURE 17

LECTURE 17
Introduction to Partial differential equation
Introduction:
In this class, the Partial differential equation will be discussed.
A partial differential equation (or briefly a PDE) is a mathematical equation that involves two or more independent variables, an unknown function (dependent on those variables), and partial derivatives of the unknown function with respect to the independent variables. The order of a partial differential equation is the order of the highest derivative involved. A solution (or a particular solution) to a partial differential equation is a function that solves the equation or, in other words, turns it into an identity when substituted into the equation. A solution is called general if it contains all particular solutions of the equation concerned.
Partial differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc.

Objectives:
equip with the concepts of partial differential equations and how to solve linear Partial Differential with different methods.
introduce to some physical problems in Engineering models that result in partial differential equations
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note
  
  Lecture note File
Select section LECTURE 18

LECTURE 18
Formation of PDE:
Introduction:
In this class, the formation of PDE will be discussed.
Partial differential equations can be obtained by the elimination of arbitrary constants or by the elimination of arbitrary functions.

Objectives:
form partial differential equation by eliminating constant
understand the properties partial differential equation
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note on formation of partial differential equation
  
  Lecture note on formation of partial differential equation File
Select section Lecture 19-20

Lecture 19-20
The linear partial differential equation of order one
Introduction:
In this class, the Linear partial differential equation of order one will be discussed. In mathematics, a first-order partial differential equation is a partial differential equation that involves only the first derivatives of the unknown function of n variables. The equation takes the form
$F(x_{1},\ldots ,x_{n},u,u_{{x_{1}}},\ldots u_{{x_{n}}})=0.\,$
Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics. If a family of solutions of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations.
Objectives:
to know about the method of characteristics
solve by using the Lagrange method
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note on Linear partial differential equation of first order
  
  Lecture note on Linear partial differential equation of first order File
Select section Lecture 21

Lecture 21
The linear partial differential equation of order two
Introduction:
In this class, the Linear partial differential equation of order two will be discussed. A second-order linear partial differential equation with two independent variables has the form
$Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots {\mbox{(lower order terms)}}=0,$
Objectives:
solve a different kind of problem
second-order partial differential equations can be classified as parabolic, hyperbolic, and elliptic.
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note on Linear equations of second order
  
  Lecture note on Linear equations of second order File
Select section lecture 22

lecture 22
Partial differential equation with constant coefficients
Introduction:
In this class, the Partial differential equation with constant coefficients will be discussed.
Objectives:
to solve higher order partial differential equations
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note
  
  Lecture note File
Select section Lecture 23-24

Lecture 23-24
Application of PDE
Introduction:
In this class, the application of PDE will be discussed. Partial differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc.
Objectives:
to know about real-life application of PDE
Lesson Plan:
Discussion from a lecture slide
Problem-solving
Group Work
Question and Answering
Home Work
- Select activity Lecture note
  
  Lecture note File
Select section You have to complete your all Quizzes

You have to complete your all Quizzes
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