Page 83 - CatalogNEP-PS

Some Concepts from Real Function theory. The Fundamental Existence and Uniqueness
Theorem. Dependence of Solutions on Initial Conditions and on the Function. Existence and
Uniqueness Theorems for Systems and Higher-Order equations. Introduction. Basic theory of the
Homogeneous Linear System. Further theory of the Homogeneous Linear System. The
Nonhomogeneous Linear System. Basic theory of the nth- Order Homogeneous Linear
Differential Equation. The nth-Order Nonhomogeneous Linear equation.
UNIT II (15 Hours)

Sturm-Liouville Problems. Orthogonality of Characteristic Functions. The Expansion of a
Function in a Series of Orthonormal Functions. The separation theorem, Sturm‘s fundamental
theorem Modification due to Picone, Conditions for Oscillatory or non-oscillatory solution, First
and Second comparison theorems. Sturm‘s Oscillation theorems. Application to Sturm Liouville
System.
UNIT III (15 Hours)

Phase Plane, Paths, and Critical Points. Critical Points and paths of Linear Systems. Critical
Points and Paths of Nonlinear Systems. Limit Cycles and Periodic Solutions. The Method of
Kryloff and Bogoliuboff.

*TUTORIAL (15 Hours (1 Hour per week))

SUGGESTED READING

1. S.L. Ross, Differential Equations, Third Edition, John Wiley & Sons, Inc.
2. E.L. Ince, Ordinary Differential Equations, Dover Publication Inc. 1956.
3. W. Boyce and R. Diprima, Elementary Differential Equations and Boundary value
Problems, 3rd Ed. New York, (1977).
4. E.A. Coddington, An Introduction to Ordinary Differential Equations, 2nd Ed. Prentice
Hall of India Pvt. Ltd., Delhi, (1974).

Math.421 Field Theory 3+1*

LEARNING OBJECTIVES:

The primary objective of this course is to introduce:
 The reducible and irreducible polynomials and their roots and identify the relations of one
field to another (known as the concept of field extension).
 The field extensions, Algebraic extensions, Normal extensions, algebraically closed fields,
splitting fields and Galosi theory
 The Galois theory which creates a bridge to move from a field to a group, and make some
remarkable observations using group theory and acquire knowledge of separable
extensions, automorphism group and fixed fields fundamental theorems of Galois theory
and algebra.

LEANING OUTCOMES:

On completion of the course, students shall be able to
 Develop the understanding about the reducible and irreducible polynomials and their roots
and identify the relations of one field to another (known as the concept of field extension).
 Attain the knowledge of field extensions, Algebraic extensions, Normal extensions,
algebraically closed fields, and Splitting fields.
 Understand the Galois theory which creates a bridge to move from a field to a group, and
make some remarkable observations using group theory and acquire knowledge of

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