Book description
Differential Equations presents the basics of differential equations, adhering to the UGC curriculum for undergraduate courses on differential equations offered by all Indian universities. With equal emphasis on theoretical and practical concepts, the book provides a balanced coverage of all topics essential to master the subject at the undergraduate level, making it an ideal classroom text. Written in lucid, easytounderstand language, the topics discussed in this studentfriendly book are amply supported by exhaustive number of problems as well as over 300 solved examples and 400 endofchapter exercises.
Table of contents
 cover
 Title Page
 Contents
 About the Author
 Preface
 1. Formation of a Differential Equation

2. Differential Equations of First Order and First Degree
 2.1 First order and first degree differential equations
 2.2 Applications of ordinary differential equations

3. Linear Differential Equations with Constant Coefficients

3.1 Introduction
 3.1.1 Linear differential equations of the second order
 3.1.2 Homogeneous equations—superposition or linearity principle
 3.1.3 Fundamental theorem for the homogeneous equation
 3.1.4 Initial value problem (IVP)
 3.1.5 Linear dependence and linear independence of solutions
 3.1.6 General solution, basis and particular solution
 3.1.7 Second order linear homogeneous equations with constant coefficients
 3.1.8 Higher order linear equations
 3.1.9 Linearly independent (L.I.) solutions
 3.1.10 Exponential shift
 3.1.11 Inverse operator D1 or
 3.1.12 General method for finding the P. I.

3.2 General solution of linear equation f(D)y = Q(x)
 Exercise 3.4
 3.2.1 Short methods for finding the particular integrals in special cases
 3.2.2 Linear equations with variable coefficients—Euler–Cauchy equations (Equidimensional equations)
 3.2.3 Legendre's linear equation
 3.2.4 Method of variation of parameters
 3.2.5 Systems of simultaneous linear differential equations with constant coefficients

3.1 Introduction
 4. Differential Equations of the First Order but not of the First Degree
 5. Linear Equation of the Second Order with Variable Coefficients

6. Integration in Series: Legendre, Bessel and Chebyshev Functions

6.1 Legendre functions
 6.1.1 Introduction
 6.1.2 Power series method of solution of linear differential equations
 6.1.3 Existence of series solutions: Method of Frobenius
 6.1.4 Legendre functions
 6.1.5 Legendre polynomials Pn(x)
 6.1.6 Generating function for Legendre polynomials Pn(x)
 6.1.7 Recurrence relations of Legendre functions
 6.1.8 Orthogonality of functions
 6.1.9 Orthogonality of Legendre polynomials Pn(x)
 6.1.10 Betrami's result
 6.1.11 Christoffel's expansion
 6.1.12 Christoffel's summation formula
 6.1.13 Laplace's first integral for Pn(x)
 6.1.14 Laplace's second integral for Pn(x)
 6.1.15 Expansion of f(x) in a series of Legendre polynomials

6.2 Bessel functions
 6.2.1 Introduction
 6.2.2 Bessel functions
 6.2.3 Bessel functions of nonintegral order p: Jp(x) and Jp(x)
 6.2.4 Bessel functions of order zero and one: J0(x), J1(x)
 6.2.5 Bessel function of second kind of order zero Y0(x)
 6.2.6 Bessel functions of integral order: Linear dependence of Jn(x) and Jn(x)
 6.2.7 Bessel functions of the second kind of order n: Jn(x): Determination of second solution Jn(x) by the method of variation of parameters
 6.2.8 Generating functions for Bessel functions
 6.2.9 Recurrence relations of Bessel functions
 6.2.10 Bessel's functions of halfintegral order
 6.2.11 Differential equation reducible to Bessel's equation
 6.2.12 Orthogonality
 6.2.13 Integrals of Bessel functions
 6.2.14 Expansion of sine and cosine in terms of Bessel functions
 6.3 Chebyshev polynomials

6.1 Legendre functions

7. Fourier Integral Transforms
 7.1 Introduction
 7.2 Integral transforms
 7.3 Fourier integral theorem
 7.4 Fourier integral in complex form
 7.5 Fourier transform of f(x)
 7.6 Finite Fourier sine transform and finite Fourier cosine transform (FFCT)
 7.7 Convolution theorem for Fourier transforms
 7.8 Properties of Fourier transform
 7.9 Parseval's identity for Fourier transforms
 7.10 Parseval's identities for Fourier sine and cosine transforms

8. Partial Differential Equations
 8.1 Introduction
 8.2 Order, linearity and homogeneity of a partial differential equation
 8.3 Origin of partial differential equation
 8.4 Formation of partial differential equation by elimination of two arbitrary constants
 8.5 Formation of partial differential equations by elimination of arbitrary functions
 8.6 Classification of firstorder partial differential equations
 8.7 Classification of solutions of firstorder partial differential equation
 8.8 Equations solvable by direct integration
 8.9 Quasilinear equations of first order
 8.10 Solution of linear, semilinear and quasilinear equations
 8.11 Nonlinear equations of first order
 8.12 Euler's method of separation of variables

8.13 Classification of secondorder partial differential equations 854
 8.13.1 Introduction
 8.13.2 Classification of equations
 8.13.3 Initial and boundary value problems and their solution
 8.13.4 Solution of onedimensional heat equation (or diffusion equation)
 8.13.5 Onedimensional wave equation
 8.13.6 Vibrating string with zero initial velocity
 8.13.7 Vibrating string with given initial velocity and zero initial displacement
 8.13.8 Vibrating string with initial displacement and initial velocity
 8.13.9 Laplace's equation or potential equation or twodimensional steadystate heat flow equation
 Acknowledgements
 Copyright
Product information
 Title: Differential Equations
 Author(s):
 Release date: January 2012
 Publisher(s): Pearson India
 ISBN: 9788131770375
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