Third Semester (B.Tech)
Question Bank
Topic – Fourier Series, Fourier Transform and Harmonic Analysis
Q1
Find the Fourier series to represent the function f (x) , given by
for 0  x  
x
f ( x)  
2  x for   x  2
1
1
1
2



..........
..

8
12 2 2 3 2
Find the Fourier transforms of
1  x 2 if x  1
f ( x)  
if x  1
0
Hence evaluate
  x cos x  sin x 
 x
 cos dx
3
0 
x
 2
Obtain a Fourier expansion for 1 cos x in the interval    x   .
Obtain the Fourier expansion
  , if    x  0
f ( x)  
and hence deduce that
0 x 
 x, if
Deduce that
Q2
Q3
Q4
2
8
Q5

1
1
2
n 1 ( 2n  1)

Find the first three harmonics for the function by the following table:-
0
f ( )
0
0.8
60
0.6

Q6
Use the integral
e
120
0.4
 y2
180
0.7
240
0.9
Q7
360
0.8
dy   , to prove that the Fourier transform of e  x

e s / 2 .
Obtain the Fourier series for the function f(x) given by
 2x
1   , if    x  0
f ( x)  
 1  2 x , if 0  x  


Hence deduce that
1 1
1
2



..........
..

8
12 3 2 5 2
2
300
1.1
2
/2
is
Q8
Q9
Q10
Q11
Q12
x
Y
Q13.
Obtain the half-range Sine series for f ( x)  2  x for 0 < x <2.
Hence deduce that
1 1 1

   ............ 
5
4
1 3
if    x  0
0,
If f ( x)  
0 x 
 sin x, if
Prove that
1 sin x 2  cos 2mx
f ( x)  
 
. Hence show that

2
 m 1 4m 2  1
1
1
1
1


 .............  (  2) .
1.3 3.3 5.7
4
Express
1
1
 4  x, if 0  x  2
f ( x)  
 x  3 , if 1  x 1

4
2
as the Fourier Series of sine terms.
Prove that 0  x   ,
 2  cos 2 x cos 4 x cos 6 x

x(  x) 
 2 

 ............
2
2
6  1
2
3

Following values of y give the displacement of a certain part for the rotation x of
the flywheel
0

2

2
4
5
3
3
3
3
1.98
2.15
2.77
.22
-0.31
1.43
1.98
Express y in Fourier Series upto the third harmonic.
Find the Fourier series for the function
 K ,    x   
f ( x)  

K , 0  x   
f(x + 2) = f(x)
and hence show that
1
Q14.
1 1 1

   ...... 
3 5 7
4
Find the Fourier transform of x exp (-x2)
Q15.
Sketch f(x) and its two periodic extensions and find the Fourier cosine series for
the function f(x), where
 2K
x

 L
f ( x)  
 2 K ( L  x)

 L
if
if




L/2
 x  L


0 x
L
L
Q16.
Find the fourier transform of x exp (-ax2), where a > 0.
Q17.
The function f(x) is given by
f ( x)
1
x ,
4
3
x
,
4


1
2
for
0 x
for
1
 x 1
2
Draw its graph and find its half range sine series

x2
2
s2
1 
is e 4 .
2
Q18.
Show that Fourier cosine transform of e
Q19.
Expand Function f(x) = x sin x as Fourier series in interval    x   and
deduce that
1
1
1
1
1



 ......    2 
1.3 3.5 5.7 7.9
4
Q20.
The turning moment T on the crank-shaft of a steam engine for the crank angle 
is as follows.

0°
15°
30°
45°
60°
75°
90°
105° 120° 135° 150° 165° 180°
T
0
2.7
5.2
7
8.1
8.3
7.9
6.8
5.5
Expand T in a series of sines upto second harmonics.
Q21.
Obtain a0, a1, bi in Fourier expansion of y from following.
4.1
2.6
1.2
0
x:
0
1
2
3
4
5
y:
9
18
24
28
26
20
Legendre’s Polynomial
Show that
Q1.
(2n + 1)x Pn = (n + 1) Pn+1 + nPn–1
Q2.
nPn = xPn'  Pn' 1
Q3.
(2n + 1) Pn = xPn'1  Pn'1
Q4.
(n+1)Pn = xPn'1  xPn'
Q5.
(1 – x2) Pn' = n(Pn-1 – xPn)
Prove that
Q6.
Q7.
1
1
 P ( x)dx  (n  1) P
0
n 1
n
1 Z 2
(0)

1  2 xZ  Z 
3
2 2
  (2n  1) Z n Pn ( x)
n 0
n
Q8.
Show that Jn (x) is the coefficient of Z in the expansion of e
Q9.
Show that
1

1
x 2 Pn 1 ( x) Pn 1 ( x)dx 
2n(2n  1)
(2n  1)( 2n  1)( 2n  3)
x
1
Z 
2
Z
BETA GAMA FUNCTION
Prove that
Q1.
1
   
2
Q2.
 1
    2 
 2
Q3.
 3 4
   

 2 3
Q4.
1


m   m    2 m1 (2m)
2 2

Q5.
n m  1, n  m m, n  1


xa
dx in terms of gama function.
ax
Q6.
Express
Q7.
Show that  ( m, n) = (m + 1, n) +  (m, n + 1)
Q8.
Show that
Q9.
Express
1
x
0
5
0
1

0
1
x m1  x n1
dx   (m, n)
1  x mn
x
0
m
(1  x m ) P dx in terms of beta function and hence evaluate
(1  x 3 )10 dx .
LAPLACE TRANSFORM
Q1.
(a)
(b)
Q2.
(a)
(b)
Find the inverse Laplace transform of
s
4
s  4a 4
Solve the initial value problem (IVP)
y’’ + ay’ – 2a2y = 0
y(o) = 6
y’(o) = 0

sin t

Using L.T. Show that 
dt 
t
2
0
Using convolution, find the inverse of
s2
s

2
 w2
Plot the 2 - periodic functions, given by f(t) =
2
(c)
t ,   t  
and find

  t ,   t  2
Its laplace transform.
Q3.
(a)
(b)
Q4.
(a)
(b)
Q5.
(a)
(b)
(c)
Q6.
(a)
(b)
s 2  2s  3
Find the inverse L.T. of
s( s  3)( s  2)
Find the L.T. of exponential function f(t) = ea+
s4
Find the inverse L.T. of
s ( s  1)( s 2  4)
find the L.T. of t2 U (t–3)
s 2  2s  3
Find the inverse L.T. of
s( s  3)( s  2)
Find the L.T. of exponential function f (t )  e at
Solve the simultaneous equation
(b 2  3) x  4 y  0
x  ( D 2  1) y  0
for
t>0, give that
dy
dx
x y
 0 and
 2at
when t = 0
dt
dt
e  t sin t
Find the L.T. of
t
Solve, using L.T. technique, the difference equation
d2y
dy
 3  2 y  4 x  e 3s
2
dx
dx
Where y(0) = 1
&
y’(0) = –1


 s3 
s
(ii)
L1  4
L 2
4 
2 2 
s  a 
 (s  a ) 
(b)
Solve the following boundary value problem using the L.T.
y" (t )  9 y (t )  cos 2t
 
y (0)  1, y   1
2
2  t 2 , 0  t  2

Q8
(a)
Find (i)
L f (t ), where f(t) – 6, 2  t  3
2t  5, 3  t  

Q7.
(a)
Find
(i)
 3 s 1

L1  2 2
e 3s 
 s ( s  4)

(b)
State and prove the second shifting theorem of L.T.
(a)
Solve y"ty' y  1
y(0) = 1
y’(0) = 2 using L.T.
(b)
Solve, y” + y’– 2y = 2 sin t cos t, if 0 < t < 2
&
3 sin2t – cos 2t, if t > 2;
y(0) = 1, y’(0), y’(0) = 0
(c)
Using convolution theorem, find the value of
 1 
L1 

s s  4 
 e 4 3 s 
(a)
Evaluate L1 
5/ 2 
 (s  4) 
t 2 for 0  t  1
(b)
A function f(t) is given by f(t) = 
4t for t  1
Express it in the terms of unit step function and find its L.T.
Evaluate the following
 sin at 
 cos at 
L
(a)
 . Does L
 exist?
 t 
 t 
s  1

L1  log
(b)

s 

(c)
Lsin 2t U (t   )
(a)
Using L. T. show that
1
 s2  b 
cos at  cos bt
1
dt

log
 2
0
2 
t
2s
s  a 
(b)
For the periodic function f(t) of period 4, defined by
0t 2
3t ,
f (t )  
find L[f(t)]
2t 4
6,
(ii)
Q9.
Q10
Q11.
Q12.
(c)
Q14.
(a)
(b)
Q15.
(a)
(b)
Q16.
Using L.T. solve the following yn – 3y’+ 2y = 4e2t under conditions
y(0) = -3 and y’(0) = 5
Find the L.T. of f(t) defined as f(t) = |t – 1| + |t + 1|, t  0
Solve d 2 y(dx 2  4 y  U ( x  2)
Where U is unit step function, y(0) = 0, y’(0) = 1
Find the L.T. of the function
0t c
t ,
f(t) = 
c  t  2c
(2c  t )' ,

 t sin t  1
dt   cot 1 ( s  1)
Show that L   e'
t
0
 s
Find the Laplace transform of f(t) defined as
t
f (t )  
5
Q17.

,
,
0t 4
t4
Find the laplace transform of

Sin wt
f (t )  
0

0t 

w

2
t 
w
w
Q18.
Find the Laplace transform of t 2 e t sin 4t
Q19.
sin 3t 

Find L e  4t

t 

Q20.
Find inverse Laplace transform of
S2 3
S ( S 2  a)

s
2
S e  e  s
in terms of unit step function.
S2  2
Q21.
Find inverse Laplace transform of
Q22.
State and prove convolution theorem for Laplace transform.
Q23.
Find the inverse Laplace transform of
3S  5 2
S2 8
Q24.
Solve
d2y
dy
 2  5 y  e  x sin x, where y (0) = 0, y’ (0) = 1
2
dx
dx
PARTIAL DIFFERENTIAL EQUATION AND APPLICATIONS
Q1.Form the partial differential equation by eliminating the arbitrary function f and 
from the equation z  f  y    (xy) .
 x
Q2
Form the partial differential equation by eliminating the arbitrary function f from
f ( xy  z 2 , x  y  z )  0
Q3
Form the partial differential equation from
lx  my  nz   ( x 2  y 2  z 2 )
Q4
Solve p  x 2  q  y 2
yz
zx
x y
Q5
Solve
p
q
yz
zx
xy
Q6
Solve
( z 2  2 yz  y 2 ) p  ( xy  zx)q  xy  zx
Q7
Using the method of separation of variables, solve the partial diff. eqn.
u u
4 
 3u given that u(0,y)  4e  y  e 5 y
x y
Q8
A thin uniform tightly stretched vibrating string fixed at the points x=0, & x=l
2
2 y
2  y
satisfies
the
equations
subject
to
the
initial

c
t 2
x 2
x
conditions y ( x,0)  y 0 sin 3 , Find the displacement y(x,t) at any x and any time
l
t.
2 y
2 y
 4 2 , given that y(0,t)=0, y(5,t)=0,
Q9
Solve the boundary value problem
t 2
x
 y 
y(x,0)=0 &    5 sin x
 t  x 0
Q10 The points of trisection of a string are pulled aside through the same distance d on
opposite position of equilibrium and the string is released from rest. Find the
expression for the displacement f the string at subsequent time & show that the
mid-point of the string always remain at rest.
Q11 A tightly stretched string of length l is attached at x=0 & at x=l. Find the
expression for y the displacement of the string at a distance x, given that
2x
y  A sin
at t = 0.
l
Q12 A rod of length l with insulated sides is initially at a uniform temperature u. its
ends are suddenly cooled to 0oC and are kept at that temperature. Prove that the
temperature function u(x,t) is given by

nx
c 2 2 n 2 t
u ( x, t )   bn sin
.e  (
) where bn is determined from the equation.
l
l2
n 1
Q13 Find the deflection u(x,y,t) of a rectangular membrane (0<x<1, 1<y<2) whose
boundary is fixed, given that it starts from rest and u(x,y,0)=xy(1-x)(2-y).
Q14
The points of trisection of a string of length L are parallel aside through a distance
b on opposite sides of the position of equilibrium and the string is released. Show
9b  1
2m
2mx
2mct
y ( x, t )  2  2 sin
sin
. cos
that
.
3
L
L
 m 1 m
Also show that the mid- point of the string always remains at rest
Q15. The temperature is in a semi-infinite rod is determined by
u
 2u
 C2 2  , 0  x  
t
x
with conditions
(i)
u = 0 when t = 0, x > 0
(ii)
u
 u when x = 0
x
(iii)
(Partial derivatives of u)  0 as x  . Determine the temperature
formula.
Q16.
Solve the differential equation
u  2u
, subject to the condition

t x 2
x , 0  x  1
 u 
   0 , u ( x,0)  
 x  x0
0 , x  1
and u(x,t) is bounded where x > 0, t > 0
Q17.
Solve the Laplace equation
 2u  2u

 0 , which satisfies the conditions:
x 2 y 2
u(0,y) = u(1, y) = c, u (x, 0) = 0
and
Q18.
 nx 
u(x, a) = sin 

 l 
From a differential equation by eliminating the arbitrary constants a & b from
Z = (x – a)2 + (y – b)2
Q19.
Find the differential equation of all planes which are at a constant distance ‘a’
from origin.
Q20.
Form a partial differential equation by eliminating functions f and F from
Z = f(x + iy) + F (x – iy)
Q21.
A string is stretched and fastened to two points ‘l’ apart Motion is started by


displacing the string in the form y  K l x  x 2 from which it is released at time
t = 0. Find the displacement of any point on the string at a distance x from one
end at time t.
Form a partial differential equation
Q22.
x2 + y2 = (z – c)2 tan2 a
Q23.
f(xy + z2, x + y + z) = 0
Q24.
x2 / a2 + y2 / b2 + z2 / c2 =1
BESSEL’S FUNCTION
Bessel’s Equation & Bessel Functions
Q1.
Prove J  n ( x)  (1) n J n ( x)
Q2.
Prove
Q3.
Prove
Q4.
2
sin x  J1/ 2
x

d n
x J n ( x)   x  n J n 1 ( x)
dx
(ii)
J n' ( x)   J n 1 ( x) 
(iii)
J n' ( x) 
(iv)
2n
J n ( x)  J n 1 ( x)  J n 1 ( x)
x
Prove J 5 / 2 
Q6.
2
cos x  J 1/ 2
x

(i)
n
J n ( x)
x
1
J n1 ( x)  J n1 ( x)
2

2 3  x2
3
 2 sin x  cos x
 x x
x

(1) m
1 d 
n
Show that 
 x J n ( x)  n  m J n  m ( x)
x
 x dx 
4
8
Show that  J 5 ( x)dx   J 4 ( x)  J 3 ( x)  2 J 2 ( x)
x
x
m
Q5.
and



2
Q7.
Show that

 x J 1 / 2 (2 x)dx1
0
Q8.
Q9.
Q10.
Q11.
State and prove Generating function for Jn(x)
State and prove Orthogonality of Bessel function.
State and Prove BER and BEI functions.
Show that
d
xBer ' ( x)   xBei ( x)
(i)
dx
(ii)
d
xBei' ( x)   xBer ( x)
dx
 xBer
2
a
Q12.
0
Prove that

( x)  Bei 2 ( x) dx  aBer (a) Bei (a)  Bei (a)Ber ' (a)
Q13.
xJ n'  nJn  xJ n 1
Q14.
2n Jn = x (Jn–1 + Jn + 1)
Q15.
J
Q16.
4 J n"  J n2  2 J n  J n 2
Q17.
J 02  2 J 12  J 22  .......  1
3

2
2
cos x 
 sin x 

x 
x 
=


Q18.
Express f ( x)  x 4  3x 3  x 2  5x  2 in terms of Laqendre Polynomials.
Q19.
Solve the equation y" 

y1
n2 
 4 x 2  2  y  0 in terms of Bessel’s functions.
x
x 
