**UNIT I ORDINARY DIFFERENTIAL EQUATIONS**

Higher order linear differential equations with constant coefficients – Method of variation of

parameters – Cauchy’s and Legendre’s linear equations – Simultaneous first order linear equations

with constant coefficients.

**UNIT II VECTOR CALCULUS**

Gradient Divergence and Curl – Directional derivative – Irrotational and solenoidal vector fields –

Vector integration – Green’s theorem in a plane, Gauss divergence theorem and stokes’ theorem

(excluding proofs) – Simple applications involving cubes and rectangular parallelpipeds.

**UNIT III ANALYTIC FUNCTIONS**

Functions of a complex variable – Analytic functions – Necessary conditions, Cauchy – Riemann

equation and Sufficient conditions (excluding proofs) – Harmonic and orthogonal properties of

analytic function – Harmonic conjugate – Construction of analytic functions – Conformal mapping : w= z+c, cz, 1/z, and bilinear transformation.

**UNIT IV COMPLEX INTEGRATION**

Complex integration – Statement and applications of Cauchy’s integral theorem and Cauchy’s

integral formula – Taylor and Laurent expansions – Singular points – Residues – Residue theorem –

Application of residue theorem to evaluate real integrals – Unit circle and semi-circular

contour(excluding poles on boundaries).

**UNIT V LAPLACE TRANSFORM**

Laplace transform – Conditions for existence – Transform of elementary functions – Basic properties

– Transform of derivatives and integrals – Transform of unit step function and impulse functions –

Transform of periodic functions.

Definition of Inverse Laplace transform as contour integral – Convolution theorem (excluding proof) –

Initial and Final value theorems – Solution of linear ODE of second order with constant coefficients

using Laplace transformation techniques.

**TOTAL : 60 PERIODS**

**TEXT BOOK:**

1. Bali N. P and Manish Goyal, “Text book of Engineering Mathematics”, 3rd Edition, Laxmi Publications (p) Ltd., (2008).

2. Grewal. B.S, “Higher Engineering Mathematics”, 40th Edition, Khanna Publications, Delhi, (2007).

**REFERENCES:**

1. Ramana B.V, “Higher Engineering Mathematics”,Tata McGraw Hill Publishing Company,

New Delhi, (2007).

2. Glyn James, “Advanced Engineering Mathematics”, 3rd Edition, Pearson Education, (2007).

3. Erwin Kreyszig, “Advanced Engineering Mathematics”, 7th Edition, Wiley India, (2007).

4. Jain R.K and Iyengar S.R.K, “Advanced Engineering Mathematics”, 3rdEdition, Narosa

Publishing House Pvt. Ltd., (2007).

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