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    Tpde r13 Aprilmay 15

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    P S HARSHITA
    This document contains a question paper for the subject "Transforms and Partial Differential Equations" for third semester B.E./B.Tech. students. The paper contains 10 short answer questions in Part A and 5 long answer questions in Part B. Students must answer all questions from both parts carrying a total of 100 marks. The questions cover topics such as partial differential equations, Fourier series, Fourier transforms, Z-transforms, and their applications. The recommended textbook for the subject is provided for reference.

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    Tpde r13 Aprilmay 15

    Uploaded by

    P S HARSHITA
    87 views4 pages
    This document contains a question paper for the subject "Transforms and Partial Differential Equations" for third semester B.E./B.Tech. students. The paper contains 10 short answer questions in Part A and 5 long answer questions in Part B. Students must answer all questions from both parts carrying a total of 100 marks. The questions cover topics such as partial differential equations, Fourier series, Fourier transforms, Z-transforms, and their applications. The recommended textbook for the subject is provided for reference.

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    This document contains a question paper for the subject "Transforms and Partial Differential Equations" for third semester B.E./B.Tech. students. The paper contains 10 short answer questions in Part A and 5 long answer questions in Part B. Students must answer all questions from both parts carrying a total of 100 marks. The questions cover topics such as partial differential equations, Fourier series, Fourier transforms, Z-transforms, and their applications. The recommended textbook for the subject is provided for reference.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    87 views4 pages

    Tpde r13 Aprilmay 15

    Uploaded by

    P S HARSHITA
    This document contains a question paper for the subject "Transforms and Partial Differential Equations" for third semester B.E./B.Tech. students. The paper contains 10 short answer questions in Part A and 5 long answer questions in Part B. Students must answer all questions from both parts carrying a total of 100 marks. The questions cover topics such as partial differential equations, Fourier series, Fourier transforms, Z-transforms, and their applications. The recommended textbook for the subject is provided for reference.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 4

    Question Paper Code: 77191 1

    B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2015


    Third Semester
    Civil Engineering
    MA 6351 – TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
    (Common to all branches except Environmental Engineering, Textile Chemistry, Textile Technology,
    Fashion Technology and Pharmaceutical Technology)
    (Regulation 2013)
    Time: Three hours Maximum: 100 marks
    Answer ALL questions
    PART A – (10 x 2 = 20 marks)

    1. Form the partial differential equation by eliminating the arbitrary constants a and b form
    log(az -1)  x  ay  b .
    Text Book Page No.: 1.79

    2. Find the complete solution of q  2 px .


    Text Book Page No.: 1.81

    3. The instantaneous current ' i ' at time t of an alternating current wave is given by
    i  I1 sin   t  1   I 3 sin  3 t   3   I 5 sin  5 t   5   ... . Find the effective value of
    the current ' i ' .
    Text Book Page No.: ----

    4. If the Fourier series of the function f ( x )  x,    x   with period 2 is given by


     sin 2 x sin 3 x sin 4 x 
    f ( x )  2  sin x     ...  , then find the sum of the series
     2 3 4 
    1 1 1
    1     ... .
    3 5 7
    Text Book Page No.: ----

    5. Classify the partial differential equation


    1  x  z
    2
    xx  2 xyz xy   1  y 2  z yy  xz x  3 x 2 yz y  2z  0 .

    Text Book Page No.: 3.5


    6. A rod 30cm long has its ends A and B kept at 20 C and 80 C respectively until steady state
    condition prevail. Find this steady state temperature in the rod.

    Text Book Page No.: 3.69

    7. If the Fourier transform of f ( x ) is F  f ( x )  F ( s ) , then show that


    F  f ( x  a )  e ias F ( s ) .

    Text Book Page No.: 4.6


    1
    8. Find the Fourier sine transform of .
    x
    Text Book Page No.: 4.57

    z
    9. If Z  x(n)  X ( z ) , then show that Z  a n x( n)  X   .
    a
    Text Book Page No.: 5.4

    10. State the convolution theorem on Z - transform.


    Text Book Page No.: 5.73

    PART B – (5 x 16 = 80 marks)

       
    11. a) (i) Solve: x 2  yz p  y 2  xz q  z 2  xy .

    Text Book Page No.: 1.111

     
    (ii) Solve: D2  3 DD  2 D 2 z  (2  4 x )e x  2 y .

    Text Book Page No.: 1.169

    (Or)

    b) (i) Obtain the complete solution of p2  x 2 y 2q 2  x 2 z 2 .

    Text Book Page No.: 1.73

    (ii) Solve z  px  qy  p2q 2 and obtain its singular solution.

    Text Book Page No.: 1.44


     x, 0 x /2
    12. a) (i) Find the half-range sine series of f ( x )   . Hence deduce the sum of
     - x ,  / 2  x  

    1
    the series  (2n  1)
    n 1
    2
    .

    Text Book Page No.: 2.109

    (ii) Find the complex form of the Fourier series of f ( x )  e  x in 1  x  1 .

    Text Book Page No.: 2.117

    (Or)
    b) (i) Find the Fourier series of f ( x )  sin x in   x   of periodicity 2 .

    Text Book Page No.: 2.51

    (ii) Compute upto the first three harmonic of the Fourier series of f ( x ) given by the
    following table:

    Text Book Page No.: 2.127

    u  2u
    13. a) Solve  a 2 2 subject to the conditions: u(0,t)  0  u( ,t), t  0 ;
    t  x
     x, 0 x /2
    u( x , 0)   .
      x, / 2  x 

    Text Book Page No.: 3.53

    (Or)
    b) A string is stretched and fastened to two points that are distance apart. Motion is started by

    displacing the string into the form y  k lx  x 2
     from which it is released at time t  0 . Find the
    displacement at any point of the string at a distance x from one end at any time t .

    Text Book Page No.: 3.25


    1 for x  a 
    sin x
    14. a) Find the Fourier transform of f ( x )  
     0 for x  a  0
    and hence evaluate 
    0
    x
    dx .


    sin 2 t 
    Using Parseval’s identity, prove that  2 dt  .
    0
    t 2

    Text Book Page No.: 4.20


    (Or)
     x2
    2
    b) (i) Show that the e is self-reciprocal under Fourier transform by finding the Fourier

    transform of e  a , a  0.
    2 2
    x

    Text Book Page No.: 4.28

    (ii) Find the Fourier cosine transform of x n1 .

    Text Book Page No.: 4.64

      and Z  1  az  .
    2
    1 1
    15. a) (i) Find Z r n cos n
    

    Text Book Page No.: 5.38


     z2 
    (ii) Using convolution theorem, find Z 1  .
      z  1 / 2  z  1 / 4  

    Text Book Page No.: 5.78

    (Or)
    b) (i) Using Z - transform, solve the difference equation x(n  2)  3 x(n  1)  2 x(n)  0
    given that x(0)  0 , x(1)  1 .

    Text Book Page No.: 5.111


     z 
    (ii) Using residue method, find Z 1  .
     z  2z  2 
    2

    Text Book Page No.: 5.65

    Text Book for Reference:


    “TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS”

    Publication: Hariganesh Publications Author: C. Ganesan

    To buy the book visit on www.hariganesh.com/textbook

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