Himachal Pradesh University (HPU) 2010 Model applied maths-II - Question Paper

B.Tech. 2^nd Semester Examination

Applied Mathematics-II

 

 

(9) (5) (4)Time Allowed: 3 Hours

Maximum Marks: 100

 

Attempt Four questions in all, selecting one question from each Section A, B and C. Section D is compulsory. Each Section carry 25 marks.

SECTION-A

 

1. (a) State and prove the Othogonality of Bessel

Functions. (9)

(b)   Show that for

n >-1. (6)

2. (a) Prove that ; where

. (6)

(b) Prove that . (9)

 

SECTION-B

3 (a) Solve Rodrigues Formula

(9)

(b) Prove that (6)

4        (a) Explain Generating Function of Legendres

Polynomial (5)

(b) Derive .

where

Hence find,,and . (10) .

SECTION-C

 

5  (a) Given thatfor , find the Fourier expression of f(x) and show that

(10)

(b) Find the Fourier series of

. (5)

6 (a) Find the Fourier transform of

.

Hence evaluate . (10)

 

(b) Find the inverse transform of . (5)

 

 

 

SECTION-D

 

7        Evaluate

                                                                          i.     

                                                                        ii.      ; n is any real >-1.

                                                                      iii.     

                                                                      iv.     

                                                                        v.      (15)

 

8        Prove That

(1)

(2) (15)

 

SECTION-E

 

(a)   For Fourier series , defines

(1) Dirichlets conditions.

(2)   Eulers formula for Fourier series.

(3)   Periodic functions.

(4)   Expansion of an even and odd function.

(5)   Half-range series, period 0 to . (20)

(b)   Express in terms of Legendre Polynomials. (4)

(c)    Prove that if n is any non negative integer. (4)

(d)   Prove that . (4)

(e)    Prove that (4)

(f)    Define Legendres and Bessels equations. (4)