UNIT-I
Introduction- order and Degree of DE, generation of ODE, solution of first-order Linear differential
equations, Solution of Linear equations with constant coefficients: CF (complementary function)
and PI (Particular integral), Homogeneous equation of Euler type, Cauchy’s and Legendre’s DE,
Variation of parameters, Simultaneous first order with constant co-
efficient, Applications of Differential Equations in Engineering and sciences.
UNIT-II
Introduction to vectors, Vector Functions, derivative and integral of vector functions, Gradient
divergence, curl, Solenoidal, Irrotational fields, Vector identities Directional derivatives, Line
integrals, Surface integrals, Volume Integrals, Green’s theorem Gauss divergence theorem,
verification, Stoke’s theorems Verification.
UNIT-III
Introduction, basic properties of LT, Laplace Transforms of standard functions, shifting theorem
Transforms of Derivatives and Integrals, Initial value and Final value theorems and verification of
simple problems, periodic functions, Inverse Laplace transforms using partial fractions, shifting
theorem, Convolution theorem, Applications of Laplace transforms for solving linear ordinary
differential equations up to second order with constant coefficient.
UNIT-IV
Definition of Analytic Function, Cauchy Riemann equations, Properties of analytic function,
Determination of analytic function using - Necessary and sufficient conditions for analyticity in
Cartesian and polar coordinates, Milne-Thomson’s method, Conformal mappings: magnification,
rotation, inversion, reflection, bilinear transformation, Cauchy’s integral theorem applications.
UNIT-V
Cauchy’s integral formulae and its extension with consequences, Taylor’s expansions with simple
problems, Laurent’s expansions with simple problems- Singularities, Types of Poles and Residues,
Cauchy’s residue theorem, Contour integration: Unit circle, Contour integration: semicircular
- Teacher: Drishti Agrawal