Contents – Differential Equations

Contents

Preface

About the Author

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

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

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

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)

Exercise 3.4

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

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

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 by changing the dependent variable and removing the first derivative (Reduction to normal form)

Exercise 5.2

5.3 General solution of by changing the independent variable

Exercise 5.3

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

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: 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 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

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

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

Exercise 8.9

Exercise 8.10