Ordinary Differential Equations

Unit I : Linear differential equations with constant coefficients: The second order homogeneous
equation, initial value problems for second order equations, linear dependence and independence,
formula for the Wronskian, the non-homogeneous equations of order two. 15 Lectures
 
Unit II: The homogeneous equations of order n, initial value problems for the nth order
equations, the non-homogeneous equation of nth order. Linear equations with variable
coefficients: Initial value problems for the homogeneous equations. Solutions of the
homogeneous equations, the Wronskian and linear independence. 15 Lectures
 
Unit III: Reduction of the order of a homogeneous equation, the non-homogenous equations,
homogeneous equations with analytic coefficients, the Legendre equations. Linear equations with
regular singular points: The Euler equations, second order equations with regular singular points. 15 Lectures
 
Unit IV: The Bessel equation, regular singular points at infinity. Existence and uniqueness of
solutions: The method of successive approximations, the Lipschitz condition, convergence of the
successive approximation. 15 Lectures 
 
Seminars, Tutorials, Problem solving session and group discussions on above four units
Recommended books:
1. E. A. Coddington: An introduction to ordinary differential equations. (2012) Prentice Hall
of India Pvt.Ltd. New Delhi.
Reference books:
1. G. Birkoff and G.G.Rota, Ordinary differential equations, John Willey and Sons.
2. G.F. Simmons, Differential Equations with Applications and Historical note, McGraw-Hill,
Inc. New York. (1972).
3. E.A. Coddington and Levinson, Theory of ordinary differential equations, McGraw-Hill,
New York(1955).
4. E.D. Rainvills, Elementary differential equations, The Macmillan company, New York,
(1964)
Skill Level: Beginner

Real Analysis

UNIT I: σ-algebra and Borel Sets of Real numbers, Lebesgue Outer Measure, The sigma algebra of Lebesgue
measurable sets, Outer and Inner approximation of Lebesgue Measurable Sets, Countable Additivity,
Continuity and Borel-Cantelli Lemma. 15 Lectures
UNIT II: Nonmeasurable Sets, Lebesgue Measurable Functions: Sums, Product and Composition of Measurable
Functions, Sequential Pointwise Limits and Simple Approximation, Littlewood’s Three Principles
(Statement and importance of Egoroff’s Theorem and Lusin’s Theorem) 15 Lectures
UNIT III: Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure, Lebesgue integral of a
Measurable Non-negative Function, The General Lebesgue Integral, Characterizations of Riemann and
Lebesgue Integrability. 15 Lectures
UNIT IV: Lebesgue’s Theorem (Statement Only), Functions of Bounded Variations, Jordan’s theorem, Absolutely
Continuous Functions, Integrating Derivatives: Differentiating Indefinite Integrals, The L
P Spaces:
Normed Linear Spaces, The Inequalities of Young, HÓ§lder and Minkowski, The Riesz-Fischer Theorem.
15 Lectures
Seminars, Tutorials, Problem solving session and group discussions on above four units
Recommended Books:
1. H. L. Royden, P.M. Fitzpatrick, Real Analysis, Fourth Edition, PHI Learning Pvt. Ltd., New Delhi,
2010
Reference Books:
1. G. de Barra, Measure Theory and Integration, New Age International (P) Ltd., 1981.
2. I. K. Rana, An Introduction to Measure and Integration, Narosa Book Company, 1997.
3. S. K. Berberian, Measure and Integration, McMillan, New York, 1965.
4. P. K. Jain, V. P. Gupta, Lebesgue measure and Integration, Wiley Easter Limited, 1986.
5. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Book Co, 1964.
6. P. K. Halmos, Measure Theory, Van Nostrand, 1950.

Skill Level: Beginner

Linear Algebra

Unit I: Elementary Basic concepts, Linear Independence and Bases, Dual Spaces, Annihilator of a subspace, Quotient Spaces. Inner product spaces, Linear Transformations. 15 Lectures
Unit II: The Algebra of Linear transformations, Characteristic Roots, Matrices of linear transformations, Eigen values and eigenvectors of a linear transformation, Canonical Forms: Similarity of linear transformations. 15 Lectures
Unit III: Triangular form, Nilpotent transformations, Jordan Form, Trace and transpose, Determinants. 15 Lectures
Unit IV: Hermitian, Unitary and Normal linear transformations, Bilinear Forms, Symmetric Bilinear Forms, Skew Symmetric Bilinear Forms. 15 Lectures
Seminars, Tutorials, Problem solving session and group discussions on above four units
Recommended Book(s):
1. Herstein I. N. : Topics in Algebra, 2nd Edition, Willey Eastern Limited.
2. Hoffman, Kenneth and Kunze R: Linear Algebra, Prentice Hill of India Private Limited., 1984.
Reference Books:
1. A. R. Rao and P. Bhimashankaran, Linear Algebra, Hidustan Book Agency.
2. Surjit Singh, Linear Algebra, Vikas publishing House (1997).
3. Gilbert Strang: Introduction to Linear Algebra, Wellesley-Cambridge Press

Skill Level: Beginner