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Differential Equations

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, easy-to-understand language, the topics discussed in this student-friendly book are amply supported by exhaustive number of problems as well as over 300 solved examples and 400 end-of-chapter exercises.

Table of Contents

  1. cover
  2. Title Page
  3. Contents
  4. About the Author
  5. Preface
  6. 1. Formation of a Differential Equation
    1. 1.1 Introduction
      1. 1.1.1 Differential equation
    2. 1.2 Differential equations
      1. 1.2.1 Formation of a differential equation
      2. 1.2.2 Solution of a differential equation
        1. Exercise 1.1
  7. 2. Differential Equations of First Order and First Degree
    1. 2.1 First order and first degree differential equations
      1. 2.1.1 Variable separable equation
        1. Exercise 2.1
      2. 2.1.2 Homogeneous equations
        1. Exercise 2.2
      3. 2.1.3 Non-homogeneous equations
        1. Exercise 2.3
      4. 2.1.4 Exact equations
        1. Exercise 2.4
      5. 2.1.5 Inexact equation—Reducible to exact equation by integrating factors
        1. Exercise 2.5
        2. Exercise 2.6
      6. 2.1.6 Linear equations
        1. Exercise 2.7
      7. 2.1.7 Bernoulli's equation
        1. Exercise 2.8
    2. 2.2 Applications of ordinary differential equations
      1. Exercise 2.9
      2. 2.2.1 Geometrical applications
        1. Exercise 2.10
  8. 3. Linear Differential Equations with Constant Coefficients
    1. 3.1 Introduction
      1. 3.1.1 Linear differential equations of the second order
      2. 3.1.2 Homogeneous equations—superposition or linearity principle
      3. 3.1.3 Fundamental theorem for the homogeneous equation
      4. 3.1.4 Initial value problem (IVP)
      5. 3.1.5 Linear dependence and linear independence of solutions
      6. 3.1.6 General solution, basis and particular solution
      7. 3.1.7 Second order linear homogeneous equations with constant coefficients
        1. Exercise 3.1
      8. 3.1.8 Higher order linear equations
      9. 3.1.9 Linearly independent (L.I.) solutions
      10. 3.1.10 Exponential shift
        1. Exercise 3.2
      11. 3.1.11 Inverse operator D-1 or
      12. 3.1.12 General method for finding the P. I.
        1. Exercise 3.3
    2. 3.2 General solution of linear equation f(D)y = Q(x)
      1. Exercise 3.4
      2. 3.2.1 Short methods for finding the particular integrals in special cases
        1. Exercise 3.5
        2. Exercise 3.6
        3. Exercise 3.7
        4. Exercise 3.8
        5. Exercise 3.9
      3. 3.2.2 Linear equations with variable coefficients—Euler–Cauchy equations (Equidimensional equations)
        1. Exercise 3.10
      4. 3.2.3 Legendre's linear equation
        1. Exercise 3.11
      5. 3.2.4 Method of variation of parameters
        1. Exercise 3.12
      6. 3.2.5 Systems of simultaneous linear differential equations with constant coefficients
        1. Exercise 3.13
  9. 4. Differential Equations of the First Order but not of the First Degree
    1. 4.1 Equations solvable for p
      1. Exercise 4.1
    2. 4.2 Equations solvable for y
      1. Exercise 4.2
    3. 4.3 Equations solvable for x
      1. Exercise 4.3
  10. 5. Linear Equation of the Second Order with Variable Coefficients
    1. 5.1 To find the integral in C.F. by inspection, i.e. to find a solution of
      1. Exercise 5.1
    2. 5.2 General solution of by changing the dependent variable and removing the first derivative (Reduction to normal form)
      1. Exercise 5.2
    3. 5.3 General solution of by changing the independent variable
      1. Exercise 5.3
  11. 6. Integration in Series: Legendre, Bessel and Chebyshev Functions
    1. 6.1 Legendre functions
      1. 6.1.1 Introduction
      2. 6.1.2 Power series method of solution of linear differential equations
      3. 6.1.3 Existence of series solutions: Method of Frobenius
      4. 6.1.4 Legendre functions
      5. 6.1.5 Legendre polynomials Pn(x)
      6. 6.1.6 Generating function for Legendre polynomials Pn(x)
      7. 6.1.7 Recurrence relations of Legendre functions
      8. 6.1.8 Orthogonality of functions
      9. 6.1.9 Orthogonality of Legendre polynomials Pn(x)
      10. 6.1.10 Betrami's result
      11. 6.1.11 Christoffel's expansion
      12. 6.1.12 Christoffel's summation formula
      13. 6.1.13 Laplace's first integral for Pn(x)
      14. 6.1.14 Laplace's second integral for Pn(x)
      15. 6.1.15 Expansion of f(x) in a series of Legendre polynomials
        1. Exercise 6.1
    2. 6.2 Bessel functions
      1. 6.2.1 Introduction
      2. 6.2.2 Bessel functions
      3. 6.2.3 Bessel functions of non-integral order p: Jp(x) and J-p(x)
      4. 6.2.4 Bessel functions of order zero and one: J0(x), J1(x)
      5. 6.2.5 Bessel function of second kind of order zero Y0(x)
      6. 6.2.6 Bessel functions of integral order: Linear dependence of Jn(x) and J-n(x)
      7. 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
      8. 6.2.8 Generating functions for Bessel functions
      9. 6.2.9 Recurrence relations of Bessel functions
      10. 6.2.10 Bessel's functions of half-integral order
      11. 6.2.11 Differential equation reducible to Bessel's equation
      12. 6.2.12 Orthogonality
      13. 6.2.13 Integrals of Bessel functions
      14. 6.2.14 Expansion of sine and cosine in terms of Bessel functions
        1. Exercise 6.2
    3. 6.3 Chebyshev polynomials
      1. Exercise 6.3
  12. 7. Fourier Integral Transforms
    1. 7.1 Introduction
    2. 7.2 Integral transforms
      1. 7.2.1 Laplace transform
      2. 7.2.2 Fourier transform
    3. 7.3 Fourier integral theorem
      1. 7.3.1 Fourier sine and cosine integrals (FSI's and FCI's)
    4. 7.4 Fourier integral in complex form
      1. 7.4.1 Fourier integral representation of a function
    5. 7.5 Fourier transform of f(x)
      1. 7.5.1 Fourier sine transform (FST) and Fourier cosine transform (FCT)
    6. 7.6 Finite Fourier sine transform and finite Fourier cosine transform (FFCT)
      1. 7.6.1 FT, FST and FCT alternative definitions
    7. 7.7 Convolution theorem for Fourier transforms
      1. 7.7.1 Convolution
      2. 7.7.2 Convolution theorem
      3. 7.7.3 Relation between Laplace and Fourier transforms
    8. 7.8 Properties of Fourier transform
      1. 7.8.1 Linearity property
      2. 7.8.2 Change of scale property or damping rule
      3. 7.8.3 Shifting property
      4. 7.8.4 Modulation theorem
        1. Exercise 7.1
    9. 7.9 Parseval's identity for Fourier transforms
    10. 7.10 Parseval's identities for Fourier sine and cosine transforms
      1. Exercise 7.2
  13. 8. Partial Differential Equations
    1. 8.1 Introduction
    2. 8.2 Order, linearity and homogeneity of a partial differential equation
      1. 8.2.1 Order
      2. 8.2.2 Linearity
      3. 8.2.3 Homogeneity
    3. 8.3 Origin of partial differential equation
    4. 8.4 Formation of partial differential equation by elimination of two arbitrary constants
      1. Exercise 8.1
    5. 8.5 Formation of partial differential equations by elimination of arbitrary functions
      1. Exercise 8.2
    6. 8.6 Classification of first-order partial differential equations
      1. 8.6.1 Linear equation
      2. 8.6.2 Semi-linear equation
      3. 8.6.3 Quasi-linear equation
      4. 8.6.4 Non-linear equation
    7. 8.7 Classification of solutions of first-order partial differential equation
      1. 8.7.1 Complete integral
      2. 8.7.2 General integral
      3. 8.7.3 Particular integral
      4. 8.7.4 Singular integral
    8. 8.8 Equations solvable by direct integration
      1. Exercise 8.3
    9. 8.9 Quasi-linear equations of first order
    10. 8.10 Solution of linear, semi-linear and quasi-linear equations
      1. 8.10.1 All the variables are separable
      2. 8.10.2 Two variables are separable
      3. 8.10.3 Method of multipliers
        1. Exercise 8.4
    11. 8.11 Non-linear equations of first order
      1. Exercise 8.5
    12. 8.12 Euler's method of separation of variables
      1. Exercise 8.6
    13. 8.13 Classification of second-order partial differential equations 8-54
      1. 8.13.1 Introduction
      2. 8.13.2 Classification of equations
      3. 8.13.3 Initial and boundary value problems and their solution
      4. 8.13.4 Solution of one-dimensional heat equation (or diffusion equation)
        1. Exercise 8.7
      5. 8.13.5 One-dimensional wave equation
      6. 8.13.6 Vibrating string with zero initial velocity
      7. 8.13.7 Vibrating string with given initial velocity and zero initial displacement
      8. 8.13.8 Vibrating string with initial displacement and initial velocity
        1. Exercise 8.8
      9. 8.13.9 Laplace's equation or potential equation or two-dimensional steady-state heat flow equation
        1. Exercise 8.9
        2. Exercise 8.10
  14. Acknowledgements
  15. Copyright