NOTE: Attempt any FIVE questions selecting at least TWO questions from each section.

Section-I (5/9)

Sturm Liouville Systems

•    Some properties of Sturm-Liouville equations

•    Regular, Periodic and singular Sturm-Liouville systems and its applications

Series Solutions of Second Order Linear Differential Equations

•    Series solution near an ordinary point

•    Series solution near regular singular points

Series Solution of Some Special Differential Equations

•    Hypergeometric function F(a, b, c; x) and its evaluation

•    Series solution of Bessel equation

•    Expression for Jn(X) when n is half odd integer, Recurrence formulas for Jn(X)

•    Orthogonality of Bessel functions

•    Series solution of Legendre equation

Introduction to PDEs

•    Review of ordinary differential equation in more than one variables 

•    Linear partial differential equations (PDEs) of the first order

•    Cauchy’s problem for quasi-linear first order PDEs

PDEs of Second Order

•    PDEs of second order in two independent variables with variable coefficients

•    Cauchy’s problem for second order PDEs in two independent variables

Boundary Value Problems  

•    Laplace equation and its solution in Cartesian, Cylindrical and spherical polar coordinates

•    Dirichlet problem for a circle

•    Poisson’s integral for a circle

•    Wave equation

•    Heat equation



Section-II (4/9)

Fourier Methods

•    The Fourier transform

•    Fourier analysis of generalized functions

•    The Laplace transform

Green’s Functions and Transform Methods

•    Expansion for Green’s functions

•    Transform methods

•    Closed form of Green’s functions

Variational Methods

•    Euler-Lagrange equations

•    Integrand involving one, two, three and n variables

•    Necessary conditions for existence of an extremum of a functional

•    Constrained maxima and minima