Theory Examination (Semester-II) 2015-16 ENGG MATHEMATICS-II

This question paper evaluates knowledge of ordinary differential equations (ODEs), partial differential equations (PDEs), Laplace transforms, Fourier series, Legendre & Bessel functions, and series solutions.
It is divided into three sections containing short answers, medium-length analytical problems, and long theoretical/practical questions.

SECTION – A (10 × 2 = 20 Marks)

This section contains ten brief questions requiring short explanations or quick calculations.

Topics include:

Roots of auxiliary equation of a given differential equation

Order and degree of ODE

Relation involving Legendre polynomials P2(x),P1(x)P_2(x), P_1(x)P2​(x),P1​(x)

Rodrigues formula for Legendre functions

Classification of PDE (elliptic/parabolic/hyperbolic)

Inverse Laplace transform

Laplace transform of unit step function

Fourier coefficient a0a_0a0​ or ana_nan​ for piecewise function

Particular integral of a PDE using operator method

Statement of two-dimensional heat equation

This section checks core definitions, basics of transforms, and classification skills.

SECTION – B (5 × 10 = 50 Marks)

Students must attempt any five analytical and computational problems.

Topics include:

Solving simultaneous ODEs

Variation of parameters for second-order ODE

Series solution of differential equation

Fourier series expansion of piecewise functions

Convolution theorem and inverse Laplace transform

Solving Laplace equation in a rectangle using separation of variables

Bessel function identity involving integral representation

Higher-order PDE solution using operator methods

This section measures problem-solving ability, transform techniques, PDE handling, and Fourier expansion skills.

SECTION – C (2 × 15 = 30 Marks)

Students must attempt any two long-answer questions, requiring full derivations and detailed explanations.

Topics include:

Differential Equations

Solving second-order and third-order ODEs

PI and CF techniques

PDE using Charpit’s/Lagrange’s methods

Legendre & Bessel Functions

Generating functions & properties

Polynomial expansions:
(1−2xz+z2)−n=∑znPn(x)(1 - 2xz + z^2)^{-n} = \sum z^n P_n(x)(1−2xz+z2)−n=∑znPn​(x)

Laplace Transform Applications

Solving ODEs using Laplace transform method

Transform of advanced functions (cosh, powers, products)

Fourier Series

Complete Fourier expansion of functions such as f(x)=x3f(x)=x^3f(x)=x3 in (−π,π)(-π,π)(−π,π)

This section evaluates deep mathematical reasoning, derivation skills, PDE solving criteria, and transform applications.