Download Math Ed. 334

Course title: Differential Equations and Number Theory
Course No.: Math Ed. 334
Nature of the Course :Theory
Level: B.Ed
Year: Third
Full Marks : 100
Pass Marks : 35
Periods/week: 6
Total Periods: 150
Time per period : 55 minutes
1. Course Description
This is a higher course on Differential equations and Number Theo. First part is differential
equation which presents the different physical phenomena of technical, economic and social
relation in precise mathematical language. It also deals with linear and partial differential
equations. Second part is Number Theory which presents theoretical base of Number system and
their properties.
2. General Objectives
The general objectives of this course are as follows:
 To make students understand singular solution and relations which appear in the P and C
discriminant – tac locus, nodel locus and cuspidal locus.
 To provide students ability in solving linear Differential equation with variable coefficients.
 To impart knowledge to the students in solving simultaneous Differential equations.
 To equip the students with the knowledge of Total Differential equations.
 To make students understand with the nature of integration in series.
 To make students understand the partial Differential equations of the First order.
 To make students understand the linear partial Differential equations with constant
coefficients.
 To make students familiar with Monge's method.
 To make students understand number theoretic functions.
 To make students familiar with Euler's generalization of Fermat's theory.
 To make students know about primitive roots and indices.
 To make students understand the quadratic reciprocity law.
 To equip students with the knowledge of perfect number, representation of integers as sum of
squares and Fibonacci numbers.
 To make students familiar with Pell's equation.
3. Specific Objectives and Contents
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Specific objectives
Find the singular solution and interpret it
as the envelope.
Determine the relations such as Tac,
Nodal and Cuspidal relation.
Contents
Unit I : Singular Solutions
(6)
1.1 Definition
1.2 Clairaut's equation
1.3 The envelope as singular solution
1.4 Relations but not solutions
1.5 The Tac locus, Nodal Locus and Cuspidal
locus
Define linear equations of Second order Unit II : Linear Differential equations of
second order with variable
with variable coefficients.
th
coefficients
(12)
Define linear equations of n order with
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variable coefficients.
Use different methods of solution.
Identify the order and type of
Simultaneous differential equation.
Find the solution of the differential
equations.
Define total differential equation.
State conditions of integrability and
exactness.
Use methods for solving the equation.
Give geometrical interpretation of the
integrable equation.
Determine orthogonal locus.
State non-integrability.
Determine the solution of differential
equation in series.
Use Frobenius method.
Define partial differential equation of first
order.
Determine the complete, particular,r
singular and general integrals.
Solve Lagrange's equation.
Derive Lagrange's solution to the Linear
equation.
State special types of equation.
Establish general method of solution.
Define Linear partial differential equation
with constant coefficient.
Solve Linear partial differential equation.
Solve
Homogeneous
and
nonHomogeneous
partial
differential
equation.
2.1 Linear equations of Second order
2.2 Linear equations of nth order
2.3 Method of removal of the first derivatives
2.4 Transformation of the equation by changing
the independent variable
2.5 Operational Method
2.6 Selection of an appropriate method
Unit III : Simultaneous Differential Equation
(6)
3.1 System
of
Simultaneous
Differential
equations
3.2 Type of simultaneous Differential equation
3.3 The general expression of integrals
Unit IV: Total Differential Equations
(6)
4.1 Define Total differential equation
4.2 Conditions for integrability
4.3 Condition for exactness
4.4 Methods for solving the equation
4.5 Geometrical interpretation of the integrable
equation
4.6 The locus Pdx+Qdy+Rdz=0 is orthogonal to
the locus
4.7 The non-integrable equation
Unit V : Integration in Series
(7)
5.1 Series solution when x=0 is an ordinary point
5.2 Series solution near a singular point
5.3 Frobenius Method
5.3.1 Cases of failure of Frobenius method
5.3.2 General theory of Frobenius
Unit VI: Partial Differential
(12)
6.1 Derivative of a partial differential equation
6.2 The complete and particular integrals
6.3 The singular integral
6.4 The general integral
6.5 Solution of Lagrange's equation
6.6 Lagrange's solution of the Linear equation
6.7 Some special types of equations
6.8 Standard I, II, III, IV
6.9 General method of solution
Unit VII : Linear Partial Differential Equation
with Constant Coefficient
(14)
7.1 Definition and Notations
7.2 Solution of Linear partial differential equation
7.3 Linear Homogeneous partial differential
equation
7.3.1 Complementary Function of a
homogeneous equation
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7.3.2 Condition with roots of auxiliary
equation
7.3.3 The particular integral
7.3.4 Method for finding particular Integral
including particular case F(a,b)=0
7.3.5 General method for particular Integral
7.4 The non-Homogeneous equation
7.4.1 The particular Integral
7.4.2 Equation reducible to homogeneous
Linear form
7.4.3 Case when Linear factors are not
possible
(12)
Use monge's method for solving the Unit VIII: Monge's Method
8.1
Monge's
method
of
solving
the
equations
equation.
Able to determine the determinants of 8.2 Determinant of Monge's form of the equation
8.3 Monge's method of Integrating
Monge's form of the equation.
8.4 Finding the equation of a surface under given
State Monge's method of integrating.
condition
Find the equation of a surface under given
condition.
To determine t & σ function and state Unit IX: Number-Theoretic functions and
Euler's Generalization of Fermat's
their characteristics.
theorem
(15)
To establish the mobins inversion
9.1
The
functions
t
and
σ
formula.
9.2 The Mobius inversion formula
To define greatest integer function.
To define Euler's phi function and 9.3 The greatest integer function
9.4 Euler's phi function
establish properties of phi-function.
9.5 Euler's Theorem
To define cryptography and apply it.
9.6 Properties of the Phi- function
9.7 Application of Cryptography
To define primitive roots for primes and Unit X : Primitive Roots, Indices and The
Quadratic Reciprocity Law
(20)
their characteristics.
10.1 The order of an integer modul u n
To establish the theory of indices.
10.2 Primitive Roots for primes
To state Euler's criterion.
To define Legendre symbol and establish 10.3 Composite numbers having primitive roots
10.4 The theory of indices
its properties.
To define Quadratic Reprocity and their 10.5 Euler's Criterion
10.6 The Legendre symbol and its properties
characteristics.
10.7 Quadratic Reciprocity
10.8 Quadratic congruences with composite
modulus
Unit XI : Perfect Numbers and The Fermat
To define perfect numbers.
conjecture
(20)
To calculate Mersenne Primes.
To define Fermat number and state its 11.1 The search for perfect numbers
11.2 Mersenne primes
characteristics.
To draw conditions for Pythagorean 11.3 Fermat Numbers
11.4 Pythagorean Triples
Triples.
11.5 The Famous 'Last Theorem'
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To draw relations about the famous 'Last
theorem'.
To define sum of two and more than two
squares.
To define Fibonacci sequence and
establish their relation.
Define Finite and Infinite continued
fraction and establish related theorems.
Determine the application of Pell's
equation.
Unit XII: Representation of Integers as sum of
squares and Fibonacci numbers (20)
12.1 Sum of Two squares
12.2 Sums of more than two squares
12.3 The Fibonacci Sequence
12.4 Identities Involving Fibonacci Numbers
12.5 Finite continued Fractions
12.6 Infinite continued Fractions
12.7 Pell's equation
4. Instructional Techniques
Because of the theoretical nature of the course, teacher-centered instructional techniques will be
dominant in the teaching learning process. The teacher will adopt the following techniques.
4.1 General Instructional Techniques
 Lecture
 Discussion
 Problem solving
4.2 Specific Instructional Techniques
Units I - VIII: Group assignment
Units IX - XII: Individual assignment
(Tasks base and presentation)
5. Evaluation
Students will be evaluated on the basis of the written classroom test in between and at the end of
the academic session, the classroom participation, presentation of the reports and other practical
activities. The scores obtained will be used only for feedback purposes. The Office of the
Controller of the Examinations will conduct the annual examination at the end of the academic
session to evaluate the students' performance. The types, number and marks of the subjective and
objective questions will be as follows.
Types of questions
Total questions
Number of questions Total
to be asked
to be answered and marks
marks allocated
Group A: Multiple choice items
20 questions
20 × 1 mark
20
Group B: Short answer questions
Group C: Long answer questions
8 with 3 'or' questions
2 with 1 'or' question
117
8 × 7 marks
2 × 12 marks
56
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6. Recommended Books and References
Recommended Books
Burton, D.M. (2004). Elementary Number Theory. New Delhi: Universal Book Service. (For
units IX to XII)
Mittal, P.K. (2003). A Text Book of Differential Equations. New Delhi: Har-Ananda (For units I
to VIII)
References
Koshy, T. (2005). Elementary number theory with applications. Delhi: Academic Press
Pokhrel, T.R. (2005). Fundamentals of number theory with application. Kathmandu: Sunlight
Publication.
Das, B.C. & Mukherjee, B.N. (2007). Integral Calculus. Culcutta: UN Dhur and Sons (Pvt) Ltd
India.
Bronson, R. (1994). Schaums Outline of Theory & Problems of Differential Equations. New
Delhi: McGran Hill, Inc.
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