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.
