Differential Equations

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...

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Detalles Bibliográficos
Autor principal:	Rukmangadachari, E. (-)
Formato:	Libro electrónico
Idioma:	Inglés
Publicado:	Noida : Pearson India 2012.
Colección:	Always learning.
Materias:	Differential equations > Textbooks. Physics > Textbooks.
Ver en Biblioteca Universitat Ramon Llull:	https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009820412806719

Tabla de Contenidos:

Cover
Contents
Preface
About the Author
Chapter 1: Formation of a Differential Equation
1.1 Introduction
1.1.1 Differential Equation
1.2 Differential equations
1.2.1 Formation of a Differential Equation
1.2.2 Solution of a Differential Equation
Exercise 1.1
Chapter 2: Differential Equations of First Order and First Degree
2.1 First order and first degree differential equations
2.1.1 Variable Separable Equation
Exercise 2.1
2.1.2 Homogeneous Equations
Exercise 2.2
2.1.3 Non-homogeneous Equations
Exercise 2.3
2.1.4 Exact Equations
Exercise 2.4
2.1.5 Inexact Equation-Reducible to Exact Equation by Integrating Factors
Exercise 2.5
Exercise 2.6
2.1.6 Linear Equations
Exercise 2.7
2.1.7 Bernoulli's Equation
Exercise 2.8
2.2 Applications of ordinary differential equations
Exercise 2.9
2.2.1 Geometrical Applications
Exercise 2.10
Chapter 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
Exercise 3.1
3.1.8 Higher Order Linear Equations
3.1.9 Linearly Independent (L.I.) Solutions
3.1.10 Exponential Shift
Exercise 3.2
3.1.11 Inverse operator D−1 or 1D
3.1.12 General Method for Finding the P. I.
Exercise 3.3
3.2 General solution of linear equation f (D) y = Q(x) 3-29
Exercise 2.4
3.2.1 Short Methods for Finding the Particular Integrals in Special Cases
Exercise 3.5
Exercise 3.6.
Exercise 3.7
Exercise 3.8
Exercise 3.9
3.2.2 Linear Equations with Variable Coefficients- Euler-Cauchy Equations (Equidimensional Equations)
Exercise 3.10
3.2.3 Legendre's Linear Equation
Exercise 3.11
3.2.4 Method of Variation of Parameters
Exercise 3.12
3.2.5 Systems of Simultaneous Linear Differential Equations with Constant Coefficients
Exercise 3.13
Chapter 4: Differential Equations of the First Order but not of the First Degree
4.1 Equations solvable for p
Exercise 4.1
4.2 Equations solvable for y
Exercise 4.2
4.3 Equations solvable for x
Exercise 4.3
Chapter 5: Linear Equation of the Second Order with Variable Coefficients
5.1 To find the integral in C.F. by inspection, i.e. to find a solution of
Exercise 5.1
5.2 General solution of R by changing the dependent variable and removing the first derivative (Reduction to normal form)
Exercise 5.2
5.3 General solution of by changing
Exercise 5.3
Chapter 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 Pa(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 (x)
6.1.14 Laplace's Second Integral for (x)
6.1.15 Expansion of f(x) in a Series of Legendre Polynomials
Exercise 6.1
6.2 Bessel functions
6.2.1 Introduction
6.2.2 Bessel Functions.
6.2.3 Bessel Functions of Non-integral Order p: Jp(x) and J-p(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 J-n(x)
6.2.7 Bessel Functions of the Second Kind of Order n: Yn(x): Determination of Second Solution Yn(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 Half-integral 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
Exercise 6.2
6.3 Chebyshev polynomials
Exercise 6.3
Chapter 7: Fourier Integral Transforms
7.1 Introduction
7.2 Integral transforms
7.2.1 Laplace Transform
7.2.2 Fourier Transform
7.3 Fourier integral theorem
7.3.1 Fourier Sine and Cosine Integrals (FSI's and FCI's)
7.4 Fourier integral in complex form
7.4.1 Fourier Integral Representation of a Function
7.5 Fourier transform of f (x)
7.5.1 Fourier Sine Transform (FST) and Fourier Cosine Transform (FCT)
7.6 Finite Fourier sine transform and finite Fourier cosine transform (FFCT)
7.6.1 FT, FST and FCT Alternative definitions
7.7 Convolution theorem for Fourier transforms
7.7.1 Convolution
7.7.2 Convolution Theorem
7.7.3 Relation between Laplace and Fourier Transforms
7.8 Properties of Fourier transform
7.8.1 Linearity Property
7.8.2 Change of Scale Property or Damping Rule
7.8.3. Shifting Property
7.8.4 Modulation Theorem
Exercise 7.1
7.9 Parseval's identity for Fourier transforms
7.10 Parseval's identities for Fourier sine and cosine transforms
Exercise 7.2.
Chapter 8: Partial Differential Equations
8.1 Introduction
8.2 Order, linearity and homogeneity of a partial differential equation
8.2.1 Order
8.2.2 Linearity
8.2.3 Homogeneity
8.3 Origin of partial differential equation
8.4 Formation of partial differential equation by elimination of two arbitrary constants
Exercise 8.1
8.5 Formation of partial differential equations by elimination of arbitrary functions
Exercise 8.2
8.6 Classification of first-order partial differential equations
8.6.1 Linear Equation
8.6.2 Semi-Linear Equation
8.6.3 Quasi-Linear Equation
8.6.4 Non-linear Equation
8.7 Classification of solutions of first-order partial differential equation
8.7.1 Complete Integral
8.7.2 General Integral
8.7.3 Particular Integral
8.7.4 Singular Integral
8.8 Equations solvable by direct integration
Exercise 8.3
8.9 Quasi-linear equations of first order
8.10 Solution of linear, semi-linear and quasi-linear equations
8.10.1 All the Variables are Separable
8.10.2 Two Variables are Separable
8.10.3 Method of Multipliers
Exercise 8.4
8.11 Non-linear equations of first order
Exercise 8.5
8.12 Euler's method of separation of variables
Exercise 8.6
8.13 Classification of second-order partial differential equations
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 One-dimensional Heat Equation (or diffusion equation)
Exercise 8.7
8.13.5 One-dimensional 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
Exercise 8.8.
8.13.9 Laplace's equation or potential equation or two-dimensional steady-state heat flow equation equation
Exercise 8.9
Exercise 8.10
Index.

Differential Equations

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