**Mathematics II**

Code: M201

Credits: 4

Note 1: The whole syllabus has been divided into five modules.

Note 2: Structure of the question paper

There will be three groups in the question paper. In Group A, there will be one set of multiple choice type questions spreading the entire syllabus from which 10 questions (each carrying one mark) are to be answered. From Group B, three questions (each carrying 5 marks) are to be answered out of a set of questions covering all the three modules. Three questions (each carrying 15 marks) are to be answered from Group C. Each question of Group C will have three parts covering not more than two topics (marked in bold italics faces). Sufficient questions should to be set covering all modules.

**Module I**

Ordinary differential equations (ODE)- First order and first degree: Exact equations, Necessary and sufficient condition of exactness of a first order and first degree ODE (statement only), Rules for finding Integrating factors, Linear equation, Bernoulli’s equation. General solution of ODE of first order and higher degree (different forms with special reference to Clairaut’s equation).

**Module II**

ODE- Higher order and first degree: General linear ODE of order two with constant coefficients, C.F. & P.I., D-operator methods for finding P.I., Method of variation of parameters, Cauchy-Euler equations,Solution of simultaneous linear differential equations.

**Module III**

Basics of Graph Theory: Graphs, Digraphs, Weighted graph, Connected and disconnected graphs, Complement of a graph, Regular graph, Complete graph, Subgraph,; Walks, Paths, Circuits, Euler Graph, Cut sets and cut vertices, Matrix representation of a graph, Adjacency and incidence matrices of a graph, Graph isomorphism, Bipartite graph.

**Module IV**

Tree: Definition and properties, Binary tree, Spanning tree of a graph, Minimal spanning tree, properties of trees, Algorithms: Dijkstra’s Algorithm for shortest path problem, Determination of minimal spanning tree using DFS, BFS, Kruskal’s and Prim’s algorithms.

**Module V**

Improper Integral: Basic ideas of improper integrals, working knowledge of Beta and Gamma functions (convergence to be assumed) and their interrelations.

Laplace Transform (LT): Definition and existence of LT, LT of elementary functions, First and second shifting properties, Change of scale property; LT of L.T. of [tex]\frac{{f(t)}}{t}[/tex]

of [tex]{t^n}f(t)[/tex] , LT of derivatives of *f (t )* ,[tex]\int f (u)du[/tex] . Evaluation of improper integrals using LT, LT of periodic and step functions, Inverse LT: Definition and its properties; Convolution Theorem (statement only) and its application to the evaluation of inverse LT, Solution of linear ODE with constant coefficients (initial value problem) using

LT.

**Suggested Reference Books:**

- 1. Advanced Engineering Mathematics, Erwin Kreyszig, (Wiley Eastern)
- 2. Graph Theory: V. K. Balakrishnan, (Schaum’s Outline, TMH)
- 3. A first course at Graph Theory: J. Clark and D. A. Holton (Allied Publishers LTD)
- 4. Introduction to Graph Theory: D. B. West (Prentice-Hall of India)
- 5. Graph Theory: N. Deo (Prentice-Hall of India)
- 6. Engineering Mathematics: B.S. Grewal (S. Chand & Co.)
- 7. Higher Engineering Mathematics: John Bird (4th Edition, 1st Indian Reprint 2006, Elsevier)
- 8. Calculus: Strauss, Bradley and Smith (3PrdP edition, Pearson Education)
- 9. Engineering Mathematics (Volume 2): S. S. Sastry (Prentice-Hall of India)
- 10. Advanced Engineering Mathematics, 3E: M.C. Potter, J.L. Goldberg and E.F. Abonfadel (OUP), Indian Edition
- 11. An Introduction to Differential Equations, R.K. Ghosh and K.C.Maity ( New Central Book Agency )

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