Engineering Mathematics II (Semester II) for UPTU

Cover
Contents
Preface
Syllabus
Chapter 1: Preliminaries
	1.1 Sets and Functions
	1.2 Continuous and Piecewise Continuous Functions
	1.3 Derivability of a Function and Piecewise Smooth Functions
	1.4 The Riemann Integral
	1.5 The Causal and Null Function
	1.6 Functions of Exponential Order
	1.7 Periodic Functions
	1.8 Even and Odd Functions
	1.9 Sequence and Series
	1.10 Series of Functions
	1.11 Partial Fraction Expansion of a Rational Function
	1.12 Special Functions
	1.13 The Integral Transforms
Chapter 2: Differential Equations
	2.1 Definitions and Examples
	2.2 Formulation of Differential Equation
	2.3 Solution of Differential Equation
	2.4 Differential Equations of First Order
	2.5 Separable Equations
	2.6 Homogeneous Equations
	2.7 Equations Reducible to Homogeneous Form
	2.8 Linear Differential Equations
	2.9 Equations Reducible to Linear Differential Equations
	2.10 Exact Differential Equation
	2.11 The Solution of Exact Differential Equation
	2.12 Equations Reducible to Exact Equation
	2.13 Applications of First Order and First Degree Equations
	2.14 Linear Differential Equations
	2.15 Solution of Homogeneous Linear Differential Equation with Constant Coefficients
	2.16 Complete Solution of Linear Differential Equation with Constant Coefficients
	2.17 Method of Variation of Parameters to Find Particular Integral
	2.18 Differential Equations with Variable Coefficients
	2.19 Simultaneous Linear Differential Equations with Constant Coefficients
	2.20 Applications of Linear Differential Equations
	2.21 Mass-Spring System
	2.22 Simple Pendulum
	2.23 Solution in Series
	2.24 Bessel’s Equation and Bessel’s Function
	2.25 Fourier-Bessel Expansion of a Continuous Function
	2.26 Legendre’s Equation and Legendre’s Polynomial
	2.27 Fourier–Legendre Expansion of a Function
	Exercises
Chapter 3: Partial Differential Equations
	3.1 Formulation of Partial Differential Equation
	3.2 Solutions of a Partial Differential Equation
	3.3 Non-linear Partial Differential Equations of the First Order
	3.4 Charpit’s Method
	3.5 Some Standard forms of Non-linear Equations
	3.6 Linear Partial Differential Equations with Constant Coefficients
	3.7 Classification of Second Order Linear Partial Differential Equations
	3.8 The Method of Separation of Variables
	3.9 Basic Partial Differential Equations
	3.10 Solutions of Laplace Equation
	3.11 Telephone Equations of a Transmission Line
	3.12 Miscellaneous Example
	Exercises
Chapter 4: Fourier Series
	4.1 Trigonometric Series
	4.2 Fourier (or Euler) Formulae
	4.3 Periodic Extension of a Function
	4.4 Fourier Cosine and Sine Series
	4.5 Complex Fourier Series
	4.6 Spectrum of Periodic Functions
	4.7 Properties of Fourier Coefficients
	4.8 Dirichlet’s Kernel
	4.9 Integral Expression for Partial Sums of a Fourier Series
	4.10 Fundamental Theorem (Convergence Theorem) of Fourier Series
	4.11 Applications of Fundamental Theorem of Fourier Series
	4.12 Convolution Theorem for Fourier Series
	4.13 Integration of Fourier Series
	4.14 Differentiation of Fourier Series
	4.15 Examples of Expansions of Functions in Fourier Series
	4.16 Method to Find Harmonics of Fourier Series of a Function from Tabular Values
	4.17 Signals and Systems
	4.18 Classification of Signals
	4.19 Classification of Systems
	4.20 Response of a Stable Linear Time Invariant Continuous Time System (LTC System) to a Piecewise Smooth and Periodic Input
	4.21 Application to Differential Equations
	4.22 Application to Partial Differential Equations
	4.23 Miscellaneous Examples
	Exercises
Chapter 5: Laplace Transform
	5.1 Definition and Examples of Laplace Transform
	5.2 Properties of Laplace Transforms
	5.3 Limiting Theorems
	5.4 Miscellaneous Examples
	Exercises
Chapter 6: Inverse Laplace Transform
	6.1 Definition and Examples of Inverse 
Laplace Transform
	6.2 Properties of Inverse Laplace Transform
	6.3 Partial Fractions Method to Find Inverse Laplace Transform
	6.4 Heaviside’s Expansion Theorem
	6.5 Series Method to Determine Inverse Laplace Transform
	6.6 Convolution Theorem
	6.7 Complex Inversion Formula
	6.8 Miscellaneous Examples
	Exercises
Chapter 7: Applications of Laplace Transform
	7.1 Ordinary Differential Equations
	7.2 Simultaneous Differential Equations
	7.3 Difference Equations
	7.4 Integral Equations
	7.5 Integro-Differential Equations
	7.6 Solution of Partial Differential Equation
	7.7 Evaluation of Integrals
	7.8 Miscellaneous Examples
	Exercises
Appendix: Model Question Papers
	Model Paper I
	Model Paper II
Index