Advanced Mathematics GE 604 power series and Bessel function & Legendre Polynomial

Legendre Polynomials:-In many applications the parameter n in Legendre's equation will be a non negative integer. are called Legendre polynomials. Since they areOf great practical importance, let us consider them in more detail, for this purpose we write in the form:

And may then express all the no vanishing coefficients in terms of the coefficient(c n) of the highest power of (x) of the polynomial. It is customaryTo choose c n=1 when n=0 and:

Differential Equation Legendre's

General Solution:

Legendre Polynomials:

Example (1)

Show That: Let:

Example (2)

Solution:

Example (3) Find

Solution:Let

Example (4)Using Power series to solve Legendre's Equation :-

Whit n=3 & n=5, Find Legendre Polynomials

K=0,1,2,3,

n=2,4,6

Example (5)Show That

Example (6)Find The Function: Solution:

General Solution:

Exampl (7)Solve the differential equation

Using Power Series.Solution:

By equation the coefficient to zero we see that the indicial equation is:

n=0,1,2,

n=0,1,2,

n=0,1,2,..

The Logarithmic terms drop out simplification yields:

Example (8)

Find a solution of

Which is boutded

)

10 Fall 2015