Introduction to Ordinary Differential equations

Order , Degree, Formation and Solution of ODE

Method of finding complimentary Functions

Second order ODE: Methods of finding Particular Integral

Second order ODE, Series Solution: Ordinary Points

ODE, Regular singular Points, Frobenius Method

Method of finding Particular Integral, Power series and solving first order differential equation

Solution of Second order ODE having regular singular points

Theorem on Frobenius Series solution of Second order linear ODE

Second Linearly independent solution of Second order ODE

Solution of ODE having two equal roots of indicial equations

Examples to find indicial equation and their roots

Gauss's Hyper Geometric Equations

Solution of Gauss Hyper geometric equation at x=1

Converting Equation into Hyper geometric series

Hyper geometric series

n of Hypergeometric equation at infinity

Solution of Chebyshev’s equation at ordinary point 𝒙=0

Solution of Chebyshev’s equation at regular singular point 𝒙=𝟏

Properties of Chebyshev's Polynomial

Solution of Legendre's equation at ordinary point 𝒙=𝟎

Solution of Legendre's equation at regular singular point 𝒙=𝟏

Rodrigues's Formula

Orthogonal Properties of Legendre's polynomials

Properties of Legendre's polynomial

Solution of Bessel's Equation

Gamma Function

J_n(x) and J_-n (x) are dependent

Properties of zeroes of Bessel's function

Recurrence Relation of Bessel’s Function

Properties of Bessel's Function (Part-II)

Orthogonal properties of Bessel's function

Bessel Series (Fourier Bessel Series )

Generating function for Bessel Function

Addition formula for Bessel Function

Bessel's Integral formula

Some continued fractions using Bessel function

Expansion of some function in Bessel's Series

Picard’s method of successive approximations

The Existence and Uniqueness of solutions

Remarks on Picard’s Theorem

Initial value Problem with Lipschitz condition