THEORY EXAMINATION (SEM–II) 2016-17 ENGINEERING MATHEMATICS - II

This question paper covers advanced topics of Differential Equations, Partial Differential Equations, Laplace Transforms, Fourier Series, Bessel Functions, and Legendre Polynomials.
It is divided into three sections testing conceptual understanding, analytical problem-solving, and application of mathematical methods.

SECTION – A (10 × 2 = 20 Marks)

This section contains ten short questions, each testing a fundamental concept or formula.
Topics include:

Showing a differential equation represents parabolas

Classification of PDEs

Particular integral of differential equations

Dirichlet’s conditions for Fourier series

Bessel function identity

Laplace transform shifting property

Laplace transform of sin(at)/t

One- and two-dimensional wave equations

Fourier constant term

Generating function of Legendre polynomials

This section focuses on definitions, proofs, identities, and small computations.

SECTION – B (5 × 10 = 50 Marks)

Students must attempt any five medium-length questions involving derivations and problem-solving.
Topics include:

Solving linear differential equations with operator method

Recurrence relation for Legendre polynomials

Series solution of differential equations

Solving ODEs using Laplace transforms with given initial conditions

Fourier series expansion of piecewise functions and deduced results

Solving Laplace equations with boundary conditions

Solving PDE using operator notation

Inverse Laplace transform using convolution theorem

This section tests higher-level computation, series methods, PDE solving, and transform techniques.

SECTION – C (2 × 15 = 30 Marks)

Students must attempt any two long descriptive questions.
Topics include:

Differential equations

Solving (D² – 2D + 1)y = eˣ sin x

Laplace method for integro-differential equations

Solving first-order PDE using Charpit’s method

Laplace transforms & Legendre polynomials

Laplace transform of (cos at – cos bt)/t

Expressing polynomials in Legendre form

Cosine series expansion for given interval

Bessel functions & wave equations

Identity for J₃(x) in terms of J₁(x) and J₀(x)

Solving first-order linear PDE

Displacement of a vibrating string with fixed ends and initial shape

This section evaluates analytical depth, derivation skills, and application of classical mathematical methods.