Q1:

given that and , and determine ,,and .Solution:-

Q2 : Derive Rodrigues formula By Legendre polynomials are given by

Now integrating this n times from 0 to x, we obtain

Which can be written

or

Which proves that.

Q3: prove that

Using the binomial theorem

We have

And the coefficient of in this expansion is

Which can be written as

i.e the required result thsult thus follows

Q4: Using the Rodrigues formula to determine and .Solution:-

Q5: determine Legendre polynomials

Solution:-

,,

(a)

+

From (a):-

Let m=m-2

m=n

m=n-2

Polynomial c0,c1x,c0(1-3x2),c1(x- x3)P0(x)=1c0=1P1(x)=xc1=1P2(x)=1-x2=1-3x2c0=-1/2

P3(x)=x- =x- x3c1=-3/2

P4(x)=1- +=1-10 x2+ x4c0=3/8

P5(x)= x- x3+ x5=x- x3+ x5c1=15/8

P6(x)=1-x2+ x4-

Q6: Find

Q7: 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

n=5

Q8 : find a solution In the interval -0.5