MA6351 Transforms and Partial Differential Equations QBank Jeppiaar

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University Anna University
Course Mechanical Engineering
SPR

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Sirish Pratap Ramnarine
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  • JEPPIAAR ENGINEERING COLLEGE
    Jeppiaar Nagar, Rajiv Gandhi Salai – 600 119
    DEPARTMENT OFMECHANICAL ENGINEERING
    QUESTION BANK
    III SEMESTER
    MA6351 Transforms and Partial Differential Equations
    Regulation – 2013
    Downloaded from: annauniversityedu.blogspot.com

    Page 1

  • SUBJECT : MA6351- TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
    SEMESTER / YEAR: III / II
    UNIT I - PARTIAL DIFFERENTIAL EQUATIONS
    Formation of partial differential equations – Singular integrals -- Solutions of standard types of first order
    partial differential equations - Lagrange’s linear equation -- Linear partial differential equations of second
    and higher order with constant coefficients of both homogeneous and non-homogeneous types.
    PART- A
    Q.No.
    Question
    Bloom’s
    Taxonomy
    Level
    Domain
    1.
    Form a partial differential equation by eliminating the arbitrary
    constants ‘a’ and ‘b’ from z ax
    2
    by
    2.
    Solution p=2ax, q=2by
    a= p/2x, b=q/2y therefore PDE is 2z=px+qy.
    BTL -6
    Creating
    2.
    Eliminate the arbitrary function from
    󰇛
    󰇜
    and form the
    partial differential equation MA6351 M/J 2014, N/D ‘14
    Solution: px+qy=0
    BTL -6
    Creating
    3.
    Form the PDE from 󰇛 󰇜
    󰇛 󰇜
    Solution Differentiating the given equation w.r.t x &y,
    z
    2
    [p
    2
    +q
    2
    +1]=r
    2
    .
    BTL -3
    Applying
    4.
    Find the complete integral of p+q=pq.
    Solution p=a, q=b therefore z=
    .
    BTL- 6
    Creating
    5.
    
    󰇛
    
    󰇜
    
    Form the partial differential equation by eliminating the arbitrary
    constants a, b from the relation log( az 1) x ay b. A/M’15
    Solution:
    Diff. p.w.r.t x&y,
    
    
    
    
    
    
    
    󰇛
    󰇜
    󰇛
    󰇜
    BTL -6
    Creating
    6.
    Form the PDE by eliminating the arbitrary constants a,b from the
    relation
    MA6351 MAY/JUNE 2014
    Solution: Differentiate w.r.t x and y
    p = 3ax
    2
    , q = 3by
    2
    therefore 3z = px+qy.
    BTL -6
    Creating
    7.
    Form a p.d.e. by eliminating the arbitrary constants from z =
    (2x
    2
    +a)(3y-b).
    Solution: p = 4x(3y-b), q = 3(2x
    2
    +a)
    3y – b = p /4x
    (2x
    2
    +a) = q/3. Therefore 12xz = pq.
    BTL -6
    Creating
    8.
    Form the partial differential equation by eliminating arbitrary
    function from (x
    2
    y
    2
    , z-xy) 0 [MA6351 M/J 2016]
    Solution: u = x
    2
    +y
    2
    and v= z-xy. Then = 2x, u
    y
    = 2y; v
    x
    = p –y;
    BTL -6
    Creating
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    Page 2

  • v
    y
    = q-x. 󰇻
    󰇻 
    
    9.
    Form the partial differential equation by eliminating arbitrary
    constants a and b from
    (x a)
    2
    ( y b)
    2
    z
    2
    1
    Solution: Differentiating the given equation w.r.t x &y,
    z
    2
    [p
    2
    +q
    2
    +1]=1
    BTL -6
    Creating
    10.
    Solve [D
    3
    -8DD’
    2
    -D
    2
    D’+12D’
    3
    ]z = 0 [MA6351 M/J 2017]
    Solution: The auxiliary equation is m
    3
    -m
    2
    -8m+12=0; m =2,2,-3
    The solution is z = f
    1
    (y+x)+f
    2
    (y+2x)+xf
    3
    (y+2x). [MA6351
    NOV/DEC 2014]
    BTL -3
    Applying
    11.
    Find the complete solution of q 2 px MA6351 APRIL/MAY 2015
    Solution
    Find the complete solution of q 2 px
    Solution: Let q= a then p = a/2x
    dz= pdx +qdy
    2z = alogx+2ay+2b.
    BTL -3
    Applying
    12.
    Find the complete solution of p+q=1[MA6351 NOV/DEC 2014]
    Solution Complete integral is z = ax + F(a) y +c
    Put p = a, q = 1-a. Therefore z = px + (1-a) y +c
    BTL -3
    13.
    Find the complete solution of
    MA6351 M/J 2016
    Solution Complete integral is z = ax + F(a) y +c
    Put p = a, q = a. Therefore z = px + q y +c
    BTL -3
    14.
    Solve [D
    3
    +DD’
    2
    -D
    2
    D’-D’
    3
    ]z = 0 Solution
    The auxiliary equation is m
    3
    -m
    2
    +m-1=0
    m =1,-i, i The solution is z = f
    1
    (y+x)+f
    2
    (y+ix)+f
    3
    (y-ix).
    BTL -3
    15.
    Solve (D-1)(D-D’+1)z = 0.
    Solution z =
    󰇛
    󰇜
    󰇛
    󰇜
    BTL -3
    16.
    Solve
    
    
    
    
    Solution: A.E: D[D-D’+1] = 0
    h=0, h=k-1
    z =
    󰇛
    󰇜
    󰇛
    󰇜
    BTL -3
    17.
    Solve (D
    4
    – D’
    4
    )z = 0. [MA6351 MAY/JUNE 2014]
    Solution: A.E : m
    4
    -1=0, m=±1, ±i.
    Z=C.F=f
    1
    (y+x)+f
    2
    (y-x)+f
    3
    (y+ix)+f
    4
    (y-ix).
    BTL -3
    18.
    Solve
    .01''
    2
    ZDDDD
    Solution: The given equation can be written as
    xyfeyfez
    OZDDD
    xx
    21
    1'1
    BTL -3
    19.
    Solve xdx + ydy = z.
    BTL -3
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    Page 3

  • Solution The subsidiary equation is
    z
    dz
    y
    dy
    x
    dx
    x
    dx
    =
    uyx
    y
    dy
    logloglog
    y
    x
    u
    Similarly
    z
    x
    v
    .
    20.
    Form the p.d.e. by eliminating the arbitrary constants from
    z= ax +by +ab
    Solution: z= ax+by+ab
    p = a & q=b
    The required equation z= px+qy+pq.
    BTL -3
    PART – B
    1.(a)
    Find the PDE of all planes which are at a constant distance ‘k’
    units from the origin.
    BTL -6
    Creating
    1. (b)
    Find the singular integral of
    z
    px
    qy
    1
    p
    2
    q
    2
    BTL -2
    Understandi
    ng
    2. (a)
    Form the partial differential equation by eliminating arbitrary
    function from (x
    2
    y
    2
    z
    2
    , ax by cz) 0
    BTL -6
    Creating
    2.(b)
    Find the singular integral of z px qy p
    2
    pq q
    2
    BTL -2
    Understandi
    ng
    3. (a)
    Form the partial differential equation by eliminating arbitrary
    functions f and g from z = x f(x/y) y g(x)
    BTL -6
    Creating
    3.(b)
    Find the singular integral of
    .1
    22
    qpqypxz
    MA6351 M/J 2016
    BTL -3
    Applying
    4. (a)
    Solve (D
    3
    -7DD’
    2
    -6D’
    3
    )z=sin(x+2y) .[MA6351 NOV/DEC 2014]
    BTL -3
    Applying
    4.(b)
    Form the partial differential equation by eliminating arbitrary
    function f and g from the relation
    z = x ƒ( x + t) + g( x + t)
    BTL -6
    Creating
    5. (a)
    Solve (D
    2
    -2DD’)z=x
    3
    y+e
    2x-y
    . [MA6351 NOV/DEC 2014]
    BTL -3
    Applying
    5.(b)
    Solve x(y-z)p+y(z-x)q=z(x-y). [MA6351 NOV/DEC 2014]
    BTL -3
    Applying
    6. (a)
    Find the singular integral of px+qy+p
    2
    -q
    2
    . MA6351 NOV/DEC2014
    BTL -2
    Understandi
    ng
    6.(b)
    Find the general solution of
    .
    22
    qpqpqypxz
    M/J ‘17
    BTL -3
    Applying
    7. (a)
    Find the complete solution of z
    2
    ( p
    2
    q
    2
    1) 1
    BTL -4
    Analyzing
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    Page 4

  • 7. (b)
    Find the general solution of (D
    2
    2DD'D'
    2
    )z 2cos y xsin y
    BTL -2
    Understanding
    8. (a)
    Find the general solution of (D
    2
    D'
    2
    )z x
    2
    y
    2
    BTL -2
    Understanding
    8.(b)
    Find the complete solution of p
    2
    x
    2
    y
    2
    q
    2
    x
    2
    z
    2
    A/M 2015
    BTL -2
    Understanding
    9. (a)
    Solve (D
    2
    3DD'2D'
    2
    ) z (2 4x)e
    x
    2y
    BTL -3
    Applying
    9.(b)
    Obtain the complete solution of z = px+qy+p
    2
    -q
    2
    MA6351 A/M 2015
    BTL -2
    Understanding
    10.(a)
    Solve x( y
    2
    z
    2
    ) p y(z
    2
    x
    2
    )q z(x
    2
    y
    2
    )
    BTL -3
    Applying
    10.(b)
    Solve (D
    2
    3DD'2D'
    2
    )z sin(x 5y)
    BTL -3
    Applying
    11(a)
    Solve the Lagrange’s equation (x 2z) p (2xz y)q x
    2
    y
    BTL -3
    Applying
    11(b)
    Solve (D
    2
    DD'2D'
    2
    )z 2x 3y e
    2 x
    4 y
    BTL -3
    Applying
    12(a)
    Solve (D
    2
    DD'6D'
    2
    )z y cos x
    BTL -3
    Applying
    12(b)
    Solve the partial differential equation
    󰇛
    
    󰇜
    󰇛
    
    󰇜
     MA6351 APRIL/MAY 2015, MA6351 M/J 2016
    BTL -3
    Applying
    13(a)
    Solve ( D
    2
    − DD’ − 20D’
    2
    ) z = e
    5s+y
    + sin (4x − y).
    BTL -3
    Applying
    13(b)
    Solve (2D
    2
    DD'D'
    2
    6D 3D')z xe
    y
    BTL -3
    Applying
    14(a)
    Solve (D
    2
    2DD')z x
    3
    y e
    2 x
    y
    BTL -3
    Applying
    14(b)
    Solve (D
    3
    7DD'
    2
    6D'
    3
    )z sin(x 2y)
    BTL -3
    Applying
    15(a)
    Form the PDE by eliminating the arbitrary function from the relation
    󰇡
    󰇢. [MA6351 MAY/JUNE 2014]
    BTL -6
    Creating
    15(b)
    Solve the Lagrange’s equation (x+2z)p+(2xz-y) = x
    2
    +y.[MA6351
    MAY/JUNE 2014]
    BTL -3
    Applying
    16(a)
    Solve x
    2
    p
    2
    +y
    2
    q
    2
    = z
    2
    . [MA6351 MAY/JUNE 2014]
    BTL -3
    Applying
    16(b)
    Solve (D
    2
    +DD’-6D’
    2
    )z = y cosx. [MA6351 MAY/JUNE 2014]
    BTL -3
    Applying
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    Page 5

  • UNIT II - FOURIER SERIES: Dirichlet’s conditions – General Fourier series – Odd and even functions
    –
    Half range sine series – Half range cosine series – Complex form of Fourier series – Parseval’s identity
    –
    Harmonic analysis.
    PART –A
    Q.No
    Question
    Bloom’s
    Taxonomy
    Level
    Domain
    1.
    State the Dirichlet’s conditions for a function f(x) to be expanded as
    a Fourier series. MA6351 MAY/JUNE 2014, A/M 2017
    Solution:
    (i) f(x) is periodic, single valued and finite.
    (ii) f(x) has a finite number of discontinuities in any one period
    (iii) f(x) has a finite number of maxima and minima.
    (iv) f(x) and f’(x) are piecewise continuous.
    BTL -1
    Remembering
    2.
    Find the value of a
    0
    in the Fourier series expansion of f(x)=e
    x
    in (0,2
    ). [ MA6351 MAY/JUNE 2014]
    Solution: a
    0
    =
    󰇛
    󰇜
     =
     = 0.
    BTL -1
    Remembering
    3.
    If
    󰇛
    󰇜
    
     then deduce that value
    of
    [MA6351 NOV/DEC 2014]
    Solution: Put x=0,
    BTL -1
    Remembering
    4.
    Does ƒ(x) = tanx posses a Fourier expansion?
    Solution No since tanx has infinite number of infinite
    discontinuous and not satisfying Dirichlet’s condition.
    BTL -2
    Understanding
    5.
    Determine the value of a
    n
    in the Fourier series expansion of
    ƒ
    (
    x
    )
    = x
    3
    in (- , ).
    Solution: a
    n
    = 0 since f(x) is an odd function
    BTL -5
    Evaluating
    6.
    Find the constant term in the expansion of cos
    2
    x as a Fourier
    series in the interval (- , ).
    Solution: a
    0
    = 1
    BTL -2
    Understanding
    7.
    If f(x) is an odd function defined in (-l, l). What are the values of
    a
    0
    and a
    n
    ?
    Solution: a
    n
    = 0 = a
    0
    BTL -2
    Understanding
    8.
    If the function f(x) = x in the interval 0<x<2 then find the
    constant term of the Fourier series expansion of the function f.
    Solution: a
    0
    = 4
    BTL -2
    Understanding
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    Page 6

  • 9.
    Expand f(x) =1 as a half range sine series in the interval (0,
    ).
    [MA6351 MAY/JUNE 2014]
    Solution: The sine series of f(x) in (0,
    ) is given by
    f (x) =
    nxb
    n
    n
    sin
    1
    where b
    n
    =
    0
    sin
    2
    nxdx
    = -
    0
    cos
    2
    nx
    n
    = 0 if n is even
    =
    n
    4
    if n is odd
    f(x) =
    nx
    n
    oddn
    sin
    4
    =
    4
    1
    12
    12sin
    n
    n
    xn
    .
    BTL -4
    Analyzing
    10.
    Find the value of the Fourier Series for
    f(x) = 0 -c<x<0
    = 1 0<x<c at x = 0 MA6351 M/J 2016
    Solution: f(x) at x=0 is a discontinuous point in the middle.
    f(x) at x = 0 =
    2
    )0()0( ff
    f(0-) = lim f(0 – h ) = lim 0 = 0
    h
    0 h
    0
    f(0+) = lim f(0 + h ) = lim 1 = 1
    h
    0 h
    0
    f(x) at x = 0 (0 + 1) / 2 = 1 / 2 = 0.5
    BTL -3
    Applying
    11.
    What is meant by Harmonic Analysis?
    Solution: The process of finding Euler constant for a tabular
    function is known as Harmonic Analysis.
    BTL -4
    Analyzing
    12.
    Find the constant term in the Fourier series corresponding to f (x) =
    cos
    2
    x expressed in the interval (-π,π).
    Solution: Given f(x) = cos
    2
    x =
    2
    2cos1 x
    W.K.T f(x) =
    2
    0
    a
    +
    nxbnxa
    n
    n
    n
    n
    sincos
    11
    To find a
    0
    =
    xdx
    2
    cos
    1
    =
    dx
    x
    0
    2
    2cos12
    =
    0
    2
    2sin1
    x
    x
    =
    1
    [(π + 0) – (0+0)] = 1.
    BTL -1
    Remembering
    13.
    Define Root Mean Square (or) R.M.S value of a function f(x) over
    the interval (a,b).
    Solution: The root mean square value of f(x) over the interval (a,b)
    is defined as
    BTL -3
    Applying
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    Page 7

  • R.M.S. =
    ab
    dxxf
    b
    a
    2
    )]([
    .
    14.
    Find the root mean square value of the function f(x) = x in the
    interval (0,).
    Solution: The sine series of f(x) in (a,b) is given by
    R.M.S. =
    ab
    dxxf
    b
    a
    2
    )]([
    =
    0
    ][
    0
    2
    l
    dxx
    l
    =
    3
    l
    .
    BTL -1
    Remembering
    15.
    If f(x) = 2x in the interval (0,4), then find the value of a
    2
    in the
    Fourier series expansion.
    Solution: a
    2
    =
    dxxx
    4
    0
    cos2
    4
    2
    = 0.
    BTL -5
    Evaluating
    16.
    To which value, the half range sine series corresponding to f(x) = x
    2
    expressed in the interval (0,5) converges at x = 5?.
    Solution: x = 2 is a point of discontinuity in the extremum.
    [f(x)]
    x = 5
    =
    2
    )5()0( ff
    =
    2
    ]25[]0[
    =
    2
    25
    .
    BTL -6
    Creating
    17.
    If the Fourier Series corresponding to f(x) =x in the interval (0, 2 π)
    is
    )sincos(
    2
    1
    0
    nxbnxa
    a
    n
    n
    n
    without finding the values of
    a
    0,
    a
    n
    , b
    n
    find the value of
    )(
    2
    2
    1
    2
    2
    0
    n
    n
    n
    ba
    a
    .
    [MA2211 APR/MAY 2011]
    Solution: By Parseval’s Theorem
    dxxfba
    a
    n
    n
    n
    2
    0
    2
    2
    1
    2
    2
    0
    )}([
    1
    )(
    2
    =
    dxx
    2
    0
    2
    1
    =
    2
    0
    3
    3
    1 x
    =
    2
    3
    8
    BTL -4
    Analyzing
    18.
    Obtain the first term of the Fourier series for the function f(x) =x
    2
    , -
    < x < .
    Solution: Given f(x) =x
    2
    ,is an even functionin - < x < .
    Therefore,
    .
    3
    2
    3
    22
    )(
    2
    2
    0
    3
    0
    2
    0
    x
    dxxdxxfa
    o
    BTL -1
    Remembering
    19.
    Find the co-efficient b
    n
    of the Fourier series for the function f(x) =
    xsinx in (-2, 2).
    Solution: xsinx is an even function in (-2,2) . Therefore b
    n
    = 0.
    BTL -6
    Creating
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    Page 8

  • 1.(a)
    Obtain the Fourier’s series of the function
    22
    0
    )(
    xforx
    xforx
    xf
    .
    Hence deduce that
    8
    ....
    5
    1
    3
    1
    1
    1
    2
    222
    MA6351A/M 2017
    BTL -1
    Remembering
    1.(b)
    Find the Fourier’s series of
    xxxf in )(
    And deduce that
    1
    2
    2
    8
    12
    1
    n
    n
    BTL -1
    Remembering
    2.(a)
    Find the Fourier’s series expansion of period
    l2
    for
    2
    )()( xlxf
    in the range
    )2,0( l
    . Hence deduce that
    6
    ....
    3
    1
    2
    1
    1
    1
    2
    222
    BTL -2
    Understanding
    2.(b)
    Find the Fourier series of periodicity 2 for f(x) =
    in
    Hence deduce that
    +
    +
    +……. =
    
    .
    BTL -2
    Understanding
    3.(a)
    Find the Fourier series upto second harmonic for the following
    data:
    X
    0
    1
    2
    3
    4
    5
    f(x)
    9
    18
    24
    28
    26
    20
    MA6351 APRIL/ MAY 2017
    BTL -1
    Remembering
    20.
    Find the sum of the Fourier Series for
    f(x) = x 0<x<1
    = 2 1<x<2 at x = 1.
    Solution: f(x) at x=1 is a discontinuous point in the middle.
    f(x) at x = 1 =
    2
    )1()1( ff
    f(1-) = lim f(1 – h ) = lim 1 – h = 1
    h
    0 h
    0
    f(1+) = lim f(1 + h ) = lim 2 = 2
    h
    0 h
    0
    f(x) at x = 1 (1 + 2) / 2 = 3 / 2 = 1.5
    BTL -3
    Applying
    PART – B
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    Page 9

  • 3.(b)
    Find the Fourier series of
    f(x) = 2x – x
    2
    in the interval 0 < x < 2
    BTL -1
    Remembering
    4.(a)
    Obtain the half range cosine series of the function
    l
    l
    xl
    l
    inx
    xf
    ,
    2
    2
    ,0
    .
    BTL -5
    Evaluating
    4.(b)
    Find the half range sine series of the function
    xxxf
    in
    the interval (0 , Л) .
    BTL -3
    Applying
    5.(a)
    Determine the Fourier series for the function
    .sin
    xinxxf
    MA6351 APRIL/ MAY 2015
    BTL -5
    Evaluating
    5.(b)
    Find the complex form of the Fourier series of f(x) = e
    -ax
    in (-l,l)
    MA6351 APRIL/ MAY 2017
    BTL -1
    Remembering
    6.(a)
    Find the Fourier series for
    .,sin)(
    inxxxf
    BTL -2
    Remembering
    6.(b)
    Find the Fourier series expansion of
    .22
    2
    xxxxf
    BTL -2
    Remembering
    7.(a)
    Find the Fourier series for
    .2,2/
    2/,0
    x
    x
    xf
    MA6351 APRIL/ MAY 2015
    BTL -4
    Analyzing
    7.(b)
    Find the Fourier series of ƒ(x) = x + x
    2
    in (-l, l) with period
    2l.
    BTL -3
    Applying
    8.(a)
    Find the Fourier series as far as the second harmonic to represent
    the function f(x) with period 6, given in the following table.
    X
    0
    1
    2
    3
    4
    5
    f(x)
    9
    18
    24
    28
    26
    20
    BTL -6
    Creating
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