## Contents

1. Formation of a Differential Equation

1.2.1 Formation of a differential equation

1.2.2 Solution of a differential equation

2. Differential Equations of First Order and First Degree

2.1 First order and first degree differential equations

2.1.1 Variable separable equation

2.1.3 Non-homogeneous equations

2.1.5 Inexact equation—Reducible to exact equation by integrating factors

2.2 Applications of ordinary differential equations

2.2.1 Geometrical applications

3. Linear Differential Equations with Constant Coefficients

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

3.1.11 Inverse operator *D*^{-1} or

3.1.12 General method for finding the P. I.

3.2 General solution of linear equation *f*(*D*)*y* = *Q*(*x*)

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

4. Differential Equations of the First Order but not of the First Degree

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

5.3 General solution of by changing the independent variable

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

6.1.2 Power series method of solution of linear differential equations

6.1.3 Existence of series solutions: Method of Frobenius

6.1.5 Legendre polynomials *P*_{n}(*x*)

6.1.6 Generating function for Legendre polynomials *P*_{n}(*x*)

6.1.7 Recurrence relations of Legendre functions

6.1.8 Orthogonality of functions

6.1.9 Orthogonality of Legendre polynomials *P*_{n}(*x*)

6.1.11 Christoffel's expansion

6.1.12 Christoffel's summation formula

6.1.13 Laplace's first integral for *P*_{n}(*x*)

6.1.14 Laplace's second integral for *P*_{n}(*x*)

6.1.15 Expansion of *f(x)* in a series of Legendre polynomials

6.2.3 Bessel functions of non-integral order *p*: *J*_{p}(*x*) and *J*_{-p}(*x*)

6.2.4 Bessel functions of order zero and one: *J*_{0}(*x*), *J*_{1}(*x*)

6.2.5 Bessel function of second kind of order zero *Y*_{0}(*x*)

6.2.6 Bessel functions of integral order: Linear dependence of *J*_{n}(*x*) and *J*_{-n}(*x*)

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

6.2.13 Integrals of Bessel functions

6.2.14 Expansion of sine and cosine in terms of Bessel functions

7. Fourier Integral Transforms

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

7.5.1 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

7.7.3 Relation between Laplace and Fourier transforms

7.8 Properties of Fourier transform

7.8.2 Change of scale property or damping rule

7.9 Parseval's identity for Fourier transforms

7.10 Parseval's identities for Fourier sine and cosine transforms

8. Partial Differential Equations

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 first-order partial differential equations

8.7 Classification of solutions of first-order partial differential equation

8.8 Equations solvable by direct integration

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

8.11 Non-linear equations of first order

8.12 Euler's method of separation of variables

8.13 Classification of second-order partial differential equations

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)

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

8.13.9 Laplace's equation or potential equation or two-dimensional steady-state heat flow equation