book

Differential Equations

Name: Differential Equations
Author: E. Rukmangadachari
ISBN: 9788131770375

by E. Rukmangadachari

January 2012

Beginner

468 pages

7h 34m

English

Pearson India

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cover
Title Page
Contents
About the Author
Preface
1. Formation of a Differential Equation
1.1 Introduction1.1.1 Differential equation1.2 Differential equations1.2.1 Formation of a differential equation1.2.2 Solution of a differential equationExercise 1.1
2. Differential Equations of First Order and First Degree
2.1 First order and first degree differential equations2.1.1 Variable separable equationExercise 2.12.1.2 Homogeneous equationsExercise 2.22.1.3 Non-homogeneous equationsExercise 2.32.1.4 Exact equationsExercise 2.42.1.5 Inexact equation—Reducible to exact equation by integrating factorsExercise 2.5Exercise 2.62.1.6 Linear equationsExercise 2.72.1.7 Bernoulli's equationExercise 2.82.2 Applications of ordinary differential equationsExercise 2.92.2.1 Geometrical applicationsExercise 2.10
3. Linear Differential Equations with Constant Coefficients
3.1 Introduction3.1.1 Linear differential equations of the second order3.1.2 Homogeneous equations—superposition or linearity principle3.1.3 Fundamental theorem for the homogeneous equation3.1.4 Initial value problem (IVP)3.1.5 Linear dependence and linear independence of solutions3.1.6 General solution, basis and particular solution3.1.7 Second order linear homogeneous equations with constant coefficientsExercise 3.13.1.8 Higher order linear equations3.1.9 Linearly independent (L.I.) solutions3.1.10 Exponential shiftExercise 3.23.1.11 Inverse operator D-1 or3.1.12 General method for finding the P. I.Exercise 3.33.2 General solution of linear equation f(D)y = Q(x)Exercise 3.43.2.1 Short methods for finding the particular integrals in special casesExercise 3.5Exercise 3.6Exercise 3.7Exercise 3.8Exercise 3.93.2.2 Linear equations with variable coefficients—Euler–Cauchy equations (Equidimensional equations)Exercise 3.103.2.3 Legendre's linear equationExercise 3.113.2.4 Method of variation of parametersExercise 3.123.2.5 Systems of simultaneous linear differential equations with constant coefficientsExercise 3.13
4. Differential Equations of the First Order but not of the First Degree
4.1 Equations solvable for pExercise 4.14.2 Equations solvable for yExercise 4.24.3 Equations solvable for xExercise 4.3
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 ofExercise 5.15.2 General solution of by changing the dependent variable and removing the first derivative (Reduction to normal form)Exercise 5.25.3 General solution of by changing the independent variableExercise 5.3

6. Integration in Series: Legendre, Bessel and Chebyshev Functions
6.1 Legendre functions6.1.1 Introduction6.1.2 Power series method of solution of linear differential equations6.1.3 Existence of series solutions: Method of Frobenius6.1.4 Legendre functions6.1.5 Legendre polynomials Pn(x)6.1.6 Generating function for Legendre polynomials Pn(x)6.1.7 Recurrence relations of Legendre functions6.1.8 Orthogonality of functions6.1.9 Orthogonality of Legendre polynomials Pn(x)6.1.10 Betrami's result6.1.11 Christoffel's expansion6.1.12 Christoffel's summation formula6.1.13 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 polynomialsExercise 6.16.2 Bessel functions6.2.1 Introduction6.2.2 Bessel functions6.2.3 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: Jn(x): Determination of second solution Jn(x) by the method of variation of parameters6.2.8 Generating functions for Bessel functions6.2.9 Recurrence relations of Bessel functions6.2.10 Bessel's functions of half-integral order6.2.11 Differential equation reducible to Bessel's equation6.2.12 Orthogonality6.2.13 Integrals of Bessel functions6.2.14 Expansion of sine and cosine in terms of Bessel functionsExercise 6.26.3 Chebyshev polynomialsExercise 6.3
7. Fourier Integral Transforms
7.1 Introduction7.2 Integral transforms7.2.1 Laplace transform7.2.2 Fourier transform7.3 Fourier integral theorem7.3.1 Fourier sine and cosine integrals (FSI's and FCI's)7.4 Fourier integral in complex form7.4.1 Fourier integral representation of a function7.5 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 definitions7.7 Convolution theorem for Fourier transforms7.7.1 Convolution7.7.2 Convolution theorem7.7.3 Relation between Laplace and Fourier transforms7.8 Properties of Fourier transform7.8.1 Linearity property7.8.2 Change of scale property or damping rule7.8.3 Shifting property7.8.4 Modulation theoremExercise 7.17.9 Parseval's identity for Fourier transforms7.10 Parseval's identities for Fourier sine and cosine transformsExercise 7.2
8. Partial Differential Equations
8.1 Introduction8.2 Order, linearity and homogeneity of a partial differential equation8.2.1 Order8.2.2 Linearity8.2.3 Homogeneity8.3 Origin of partial differential equation8.4 Formation of partial differential equation by elimination of two arbitrary constantsExercise 8.18.5 Formation of partial differential equations by elimination of arbitrary functionsExercise 8.28.6 Classification of first-order partial differential equations8.6.1 Linear equation8.6.2 Semi-linear equation8.6.3 Quasi-linear equation8.6.4 Non-linear equation8.7 Classification of solutions of first-order partial differential equation8.7.1 Complete integral8.7.2 General integral8.7.3 Particular integral8.7.4 Singular integral8.8 Equations solvable by direct integrationExercise 8.38.9 Quasi-linear equations of first order8.10 Solution of linear, semi-linear and quasi-linear equations8.10.1 All the variables are separable8.10.2 Two variables are separable8.10.3 Method of multipliersExercise 8.48.11 Non-linear equations of first orderExercise 8.58.12 Euler's method of separation of variablesExercise 8.68.13 Classification of second-order partial differential equations 8-548.13.1 Introduction8.13.2 Classification of equations8.13.3 Initial and boundary value problems and their solution8.13.4 Solution of one-dimensional heat equation (or diffusion equation)Exercise 8.78.13.5 One-dimensional wave equation8.13.6 Vibrating string with zero initial velocity8.13.7 Vibrating string with given initial velocity and zero initial displacement8.13.8 Vibrating string with initial displacement and initial velocityExercise 8.88.13.9 Laplace's equation or potential equation or two-dimensional steady-state heat flow equationExercise 8.9
Exercise 8.10
Acknowledgements
Copyright

Content preview from Differential Equations

8 Partial Differential Equations

8.1 INTRODUCTION

The reader is familiar with ordinary differential equations. These equations involve functions of a single variable only and their derivatives. In many problems that arise in geometry, physics, population dynamics, social sciences, medicine and engineering, one has to deal with equations containing functions of more than one variable and partial derivatives with respect to these independent variables. Such equations are called partial differential equations. Thus, a partial differential equation is an equation of the form

containing independent variables t, x, y, …, an unknown function ...

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Differential Equations

by E. Rukmangadachari

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