Post on 29-Jul-2015
ENGINEERING MATHEMATICS-III
IMPORTANT UNIVERSITY QUESTIONS
UNIT-IFOURIER SERIES
TWO MARKS
1)Determine nb in the Fourier series expansion of ( ) ( )xxf −= π2
1 in
π20 << x with period π2 . (May/June 2007)2)Define root mean square value of ( )xf in bxa << . (M/J’07)
3)If ( )ππ
π2
0
,5 0
,c o s
<<<<
=
xi f
xi fxxf and ( ) ( )π2+= xfxf for all x, find the sum of the Fourier
series of ( )xf at π=x . (Nov/Dec 2007)4)Find the value of na in the cosine series expansion of ( ) kxf = in the
interval (0,10). (Nov/Dec 2006)5)Find the root mean square value of the function ( ) xxf = in the interval ( 0,
l ). (Nov/Dec 2006)6)State Dirichlet’s conditions for a given function to expand in Fourier series.
(Nov/Dec 2003)7)If the Fourier series of the function ( ) 2xxxf += in the interval ππ <<− x is
( )∑
−−+
∞
=12
2
sin2
cos4
13 n
n nxn
nxn
π, then find the value of the infinite series
.....23
122
121
1 +++ (Nov/Dec 2003)
8)Find the Fourier sine series of the function ( ) 1=xf , π<< x0 . (Apr/May 2004)
9)If the Fourier series of the function ( )ππ
π2
0
,s i n
,0
<<<<
=
x
x
xxf is
( ) xxxx
xf sin2
1......
75
6cos
53
4cos
31
2cos21 +
+
⋅+
⋅+
⋅+−=
ππ deduce that
4
2.........
75
1
53
1
31
1 −=∞−⋅
+⋅
−⋅
π.(Apr/May 2004)
10) Does ( ) xxf tan= possess a Fourier expansion ? (Nov/Dec 2005)11) State Parseval’s Theorem on Fourier series. (Nov/Dec 2005)12) Find nb in the expansion of 2x as a Fourier series in ( )ππ ,− .
1
(Nov/Dec 2005)13) If ( )xf is an odd function defined in ( -l , l ) , what are the values of 0a
and na ? (Nov/Dec 2005)14) Find the constant term in the Fourier series corresponding to ( ) xxf 2cos=
expressed in the interval ( )ππ,− . (Oct/Nov 2002)15) To which value the Half range sine series corresponding to ( ) 2xxf =
expressed in the interval (0,2) converges to 2=x ? (Oct/Nov 2002)16) If ( ) 2xxxf += is expressed as a Fourier series in the interval (-2,2)
to which value this series converges at 2=x ? (Apr/May 2003)17) If the Fourier series corresponding to ( ) xxf = inthe interval ( )π2,0 is
( )∑ ++∞
1
0 sincos2
nxbnxaa
nn ,without finding the values of .,,0 nn baa Find the
value of ( )∑ ++∞
1
2220
2 nn baa
. (Apr/May 2003)
18) Find the Half range sine series for ( ) 2=xf in π<< x0 . (April 2001)19) If the cosine series for ( ) xxxf sin= for π<< x0 is given by
( )∑
−−−−=
∞
22 ,cos1
12cos
2
11sin nx
nxxx
n
Prove that 1+2
2........
75
1
53
1
31
1 π=
−
⋅+
⋅−
⋅ . (April 2001)
20) What do you mean by Harmonic analysis ? (Apr/May 2005)
21) In the Fourier expansion ( )π
π
π
π<<
<<−
−
+=
x
xx
x
xf0
0
,2
1
,2
1in( )ππ,− ,
find the value of nb ,the coefficient of sinnx (Apr/May 2004)22) Find na in expanding xe− as Fourier series in ( )ππ,− .(May 2006)23) State Parseval’s identity of Fourier series. (Nov/Dec 2004)
24) If ( )∑ ++=∞
1
03 sincos2
cos ntbntaa
t nn in π20 ≤≤ t ,find the sum of the series
( )∑ ++∞
1
2220
4 nn baa
.(Nov 2007)
25) The Fourier series of 2x in (0,2) and that of ( )22+x in (-2,0) are identical or not. Give reason. (Nov 2007)
26) Define the value of the Fourier series of ( )xf at a point of discontinuity (Dec 2008)27) If ( ) xsinhxf = is defined in ππ <<− x , write the values of Fourier
coefficients 0a and na . (Dec 2008)
28) If
+−+−= ...
4
x4sin
3
x3sin
2
x2sin
1
xsin2x in π<< x0 , prove that ∑ =
6
2
2n
1 π
. (Dec 2008)
29) The functions ( ) xtanxf = , ( )
=x
1sinxf cannot be expanded as a
Fourier series. Why ? (Dec 2008)
2
30) Expand the function ( ) 1xf = , π<< x0 as a series of sines. (Dec 2008)31) Find the Fourier Cosine series of ( ) x2cosxf = , π<< x0 . (Dec 2008)32) ( ) 2xxf = , 2x0 ≤≤ which one of the following is correct a)
an even function b) an odd function c) neither even nor odd (Dec 2008)33) Define root mean square value of a function ( )xf over the range (a,b) (Dec 2008) 34) Define Harmonic analysis. (Dec 2008)
35) Let ( )xf be defined in ( )π2,0 by ( )
−+
xcosx
xcos1xf π
:
: ππ
π2x
x0
<<<<
and
( ) ( )xf2xf =+ π . Find the value of ( )πf . (May/June 2009)
36) State the Dirichlet’s condition for the convergence of the Fourier series of
( )xf in [ ]π2,0 with period π2 . (May/June 2009)
SIX MARKS1) Determine the Fourier series for the function ( ) 2xxf = of period π2
in π20 << x . (May/June 2007)
2) Find the Half range cosine series for the function ( ) ( )xxxf −= π in π<< x0 .Deduce that
90
.......3
1
2
1
1
1 4
444
π=+++ . (May/June 2007) (May/June 2009)
3) Find the complex form of Fourier series for the function ( ) xexf −= in 11 <<− x . (May/June 2007)
4) Determine the Fourier series for the function ( )
<<<<−
++−
=π
πx
x
x
xxf
0
0
,1
,1
Hence deduce that 4
........5
1
3
11
π=−+− . (May/June
2007)5) By finding the Fourier cosine series for ( ) xxf = in π<< x0 , Show
that ( )∑−
=∞
=14
4
12
1
96 n n
π. (Nov/Dec 2005)
6) Find the complex form of the Fourier series of the function ( ) xexf = when ππ <<− x and ( ) ( )xfxf =+ π2 (N/D07)
7) Find the Half range cosine series of ( ) ( )2xxf −= π in the interval ( )π,0
.Hence find the sum of the series ∞++++ .......3
1
2
1
1
1444 .
(Nov/Dec 2006)8) Find the Fourier series as the second harmonic to represent the
function given un the following data: X : 0 1 2 3 4 5
Y : 9 18 24 28 26 20. (N/D 2006) 10) Find the Fourier series expansion of period and period for the
3
function ( )
−=
ll
l
in
in
xl
xxf
,2
2,0
Hence deduce the sum
of the series ( )∑−
∞
= 1412
1
n n. (Nov/Dec 2006)
11) Obtain the Fourier series of ( )xf of period 2l and defined as follows
( )
<≤≤<−
=lxl
lxxlxf
2
0
,0
, .Hence deduce that
4........
5
1
3
11
π=−+− and 8
.......5
1
3
1
1
1 2
222
π=+++ .
(Nov/Dec 2007),(Dec 2008)12) Determine the Fourier expansion of ( ) xxf = in the interval ππ <<− x .
(Apr/May 2004)13) Find the Half range cosine series for xxsin in ( )π,0 .(A/M 2004)
14) Obtain the Fourier series for the function ( ) ( )
≤≤≤≤
−=
21
10
,2
,
x
x
x
xxf
ππ
.
(Apr/May 2004)
15) Find the Fourier series of period π2 for the function ( ) ( )( )ππ
π2,
,0
2
1
i n
i nxf
=
and hence find the sum of the series ∞++++ .......5
1
3
1
1
1222 .
(Apr/May 2004), (Apr/May 2005)16) Obtain the Fourier expansion for xcos1− in ππ <<− x . (March 1996)
17) Find the Fourier series for the function ( ) ( )
<<<<
−=
21
10
,1
,
x
x
in
in
x
xxf .Deduce
that 8
.......5
1
3
1
1
1 2
222
π=∞++++ . (Nov/Dec 2005)
18) Find the Fourier series for ( ) xxf cos= in the interval ( )ππ ,− . (Nov/Dec 2005)
19) Find the Fourier series for ( ) 2xxf = in ( )ππ ,− .Hence find
∞++++ .......3
1
2
1
1
1444 . (Nov/Dec 2005) (Dec 2008)
20) Expand in Fourier series of periodicity π2 of ( ) ( )
<<<<
−=
πππ
π 2
0
,2
,
x
x
x
xxf .
4
21) Find the Fourier series expansion of the periodic function ( )xf of
period 2l defined by ( )
≤≤≤≤−
−+
=lx
xl
xl
xlxf
0
0
,
, Deduce that
( ) 812
1 2
12
π=∑−
∞
=n n (Oct/Nov 2002)
22) Obtain the half range cosine series for ( ) ( )22xxf −= in the interval (0,2). (Dec 2008)
23) Expand ( ) 2xxxf += in ( )ππ ,− as a full range Fourier series and
hence deduce the sum of the series ∑∞
=1n2n
1 (Dec 2008)
24) Expand ( ) 2x0,2xx2xf <<−= as a series of cosines. (Dec 2008)
25) Give the sine series of ( ) 1xf = in ( )π,0 and prove that 8
2
...3,12n
1 π=∞∑
(Dec 2008)26) Find the Fourier series up to second Harmonic for the data
x : 0 60 120 180 240 300 360f(x): 1 1.4 1.9 1.7 1.5 1.2 1 (Dec 2008)
27) Find the cosine series of ( ) 2xxf = in ( )π,0 (Dec 2008)
28) Find Fourier series of ( )
<≤≤<−
=lxl
lxxlxf
2
0
,0
, (Dec08) (N/D07)
29) Find the Fourier series of ( ) 2xxf = in ( )π2,0 and periodic with period
π2 . Hence deduce that ∑∞
==
1n6
2
2n
1 π (May/June 2009)
30) If a is not an integer , find the complex Fourier series of ( ) axcosxf = in ( )ππ ,− . (May/June 2009)
31) Compute the first two harmonics of the Fourier series of ( )xf given in the following table:
x : 0 3/π 3/2π π 3/4π 3/5π π2 ( )xf : 1.0 1.4 1.9 1.7 1.5 1.2 1.0 (May/June 2009)
UNIT-IIFOURIER TRANSFORM
TWO MARKS
1) Write the Fourier transform pair. (Nov/Dec 2007)
2) If ( )SFC is the Fourier cosine transform of f(x) , prove that the Fourier
cosine transform of f(ax) is
a
sF
a C
1. (Oct 2002)
3) If F(S) is Fourier transform of f(x) , write the Fourier transform of
f(x)cos(ax) in terms of F. (April 2003)
5
4) State the Convolution theorem for Fourier transforms. (M/J 2009)
(N/D2005),(Apr’03)
5) If F(S) is Fourier transform of f(x) , then find the Fourier transform of
f(x-a). (Nov/Dec 2003)
6) If ( )SFs is the Fourier Sine transform of f(x), show that
( )[ ] ( ) ( )[ ]asFasFaxxfF SSS −++=2
1cos . (Nov/Dec 2003)
7) Solve the integral equation ( ) λλ −∞
=∫ exdxxf cos0
. (April 2004)
8) Find the Fourier transform of f(x) if ( )
=0
1xf
:
:
0>><ax
ax. (April 2004)
9) Find the Fourier cosine transform of xe− . (Nov 2004)
10) Find a) ( ){ }xfxF n and b) ( )
n
n
dx
xfdF in terms of the Fourier transform of
f(x). (Nov 2004)11) State Fourier integral theorem. (April 2005) (May/June 2009)
12) Find the Fourier Sine transform of x
1. (April 2005),(Dec 2008)
13) Find the Fourier transform of 0, >− αα xe . (Nov/Dec 2005)
14) Find the Fourier cosine integral representation of ( )
=0
1xf
1
10
><<
x
x .
15) Find the Fourier Sine transform of f(x)= xe− . (May 2006)
16) Prove that ( ){ }
=a
sF
aaxfF
1 , a>0. (May 2006)
17) If ( ){ } ( )sfxfF = then give the value of ( ){ }axfF . (May 2006)
18) Find the Fourier transform of ( )
=0
1xf
1
1
><
x
x. (May 2006)
19) Find the Fourier cosine transform of f(x) defined as
( )
−=0
2 x
x
xf for
for
for
2
21
10
><<<<
x
x
x
. (Nov/Dec 2006)
20) P.T ( )[ ] ( )asFxfeF iax += where ( )[ ] ( )sFxfF = . (M/J 2007) 21) Write down the Fourier cosine transform pair formulae. (M/J 2007) 22) If ( )[ ] ( )sfxfF = prove that ( )[ ] ( )sfeaxfF ias−=− .
(N/D 2003),(Nov 2005)
23) Prove that if ( ){ } ( )sFxfF = , then ( ){ } ( )sFisaeaxfF =− (Dec 2008)
24) Find the Fourier transform of ( )xf defined by
( )
=0
1xf
,
,
otherwise
bxa <<.(Dec 2008)
6
25) Find the Fourier Cosine transform of ( )
=0
xxf
,,
ππ
≥<<
x
x0
. (Dec 2008)
26) If ( ){ } ( )sFxfF = , prove that ( ){ } ( )sF
2ds
2dxf2xF −=
. (Dec 2008)
27) Find the Fourier sine transform of x1
. (Dec 2008)
28) State Parseval’s identity on complex Fourier transforms. (Dec 2008)
29) If ( ){ } ( )sFxfF = then prove that ( ){ } ( )asFxfiaxeF += . (Dec 2008)
30) State Modulation theorem in Fourier transform. (Dec 2008)
Give a function which is self reciprocal under Fourier sine and cosine
transforms. (Dec 2008)
31) If ( ){ } ( )scFxfcF = , then prove that ( )( ) ( )( )scFds
dxxfsF −= . (Dec 2008)
SIX MARKS
1) Find the Fourier transform of f(x) =0
1 otherwise
xfor 1≤ .Hence prove that
∫ ∫∞ ∞
=
=
0 0
2
2
sinsin πdx
x
xdx
x
x. (Nov 2002),(Nov/Dec 2003)
2) Find the Fourier transform of ( )
=0
sin xxf ∞≤≤
<<x
x
ππ0
. (Nov 2002)
3) Find the Fourier cosine transform of 2,0)2(4)1(3)( ≥=−−−+ nnynyny
.Deduce that
∫∞
−=+0
82 816
2cosedx
x
x π, ∫
∞−=
+0
82 816
2sinedx
x
xx π.
4) State and prove the Convolution theorem for Fourier transforms. (Nov 2002)5) Find the Fourier transform of ( )0, >− ae xa . Deduce that (i)
( )∫∞
=+0
3222 4
1
adx
ax
π , (ii) { } ( ) 222
22
sa
asixeF xa
+=−
π .
(April 2003)
6) Find the Fourier Sine transform of 2
2x
xe− . (April 2003)
7) Find the Fourier cosine transform of 22xae− .Hence evaluate the Fourier Sine transform of 22xaxe − . (Nov/Dec 2006)
8) Find the Fourier transform of 22xae− . Hence prove that 2
2x
e− is self-
reciprocal. (May 2006), (May 2007)
7
9) Find the Fourier cosine transform of ( ) −=
0
1 2xxf
otherwise
x 10 <<. Hence
prove that ∫∞
=
−
03
.16
3
2cos
cossin πdx
x
x
xxx (April 2003)
10) Derive the Parseval’s identity for Fourier transforms. (April 2003)11) Find the Fourier Sine and cosine transform of xe− . Hence find the
Fourier Sine transform of 21 x
x
+ and Fourier cosine transform of
21
1
x+. (Nov/Dec 2003)
12) Show that Fourier transform of ( ) −=
0
22 xaxf
:
:
ax
ax
><
is
−3
acosaasin22
λλλλ
π . Hence deduce that ∫∞
=−
03
.4
cossin πdt
t
ttt
(Apr/May 2004)13) Find the Fourier Sine and cosine transform of
( )
−=0
2 x
x
xf :
:
:
2
21
10
><<<<
x
x
x
(Apr/May 2004)
14) If ( )λf is the Fourier transform of f(x) , find the Fourier transform of f(x-a) and f(ax). (Apr/May 2004)
15) Verify Parseval’s theorem of Fourier transform for the function
( )
= − xexf
0 :
:
0
0
><
x
x. (Apr/May 2004)
16) Find the Fourier transform of f(x), ( ) −=
0
1 2xxf
:
:
1
1
>≤
x
x. Hence
evaluate
(i) ∫∞
−
03 2
coscossin
dxx
x
xxx (ii) ∫
∞
=
−
0
2
3 15
cossin πds
s
sss.
(April 2005) (Dec 2008)
17) Find the Fourier Sine transform of ( )0>−
ax
e ax
. (Nov/Dec 2006)
18) Find the Fourier Sine and cosine transform of xe 2− . Hence find the
value of the following integrals (i) ( )∫∞
+022 4x
dx (ii) ( )∫
∞
+022
2
4dx
x
x.
(A.U.Model Qu)
19) Evaluate (i) ( )( )∫∞
++02222 bxax
dx (ii) ( )( )∫
∞
++022 41 xx
dxusing Fourier
transform. (Nov/Dec 2008)20) Find the Fourier Sine and cosine transform of 1−nx . (May 2006)21) Using Parseval’s identity for Fourier cosine transform of axe− evaluate
( )∫∞
+0222 xa
dx . (Nov/Dec 2007)
8
22) Find the Fourier Sine transform of ( )0, >− ae ax . Hence find [ ]axS xeF − .
Hence deduce the inversion formula. (May/June 2007)23) Find the Fourier Sine transform of f(x) defined as
( )
=0
sin xxf
where
where
ax
ax
><<0
. (Dec 2008)
24) Find the Fourier transform of ( ) −
=0
x1xf
otherwise
for
1x ≤. Hence
find the values of (i) ∫∞
0
dt4
t
tsin and (ii) ∫
∞
0
dx2
x
xsin (Dec 2008)
25) Find the finite sine and cosine transform of ( )2x
1xf
−=
π in the interval
( )π,0 . (Dec 2008)
26) Find the Fourier transform of ( ) −
=0
xaxf
,
,
ax
ax
><
. (Dec 2008)
27) Evaluate ∫∞
+
+02x2b2x2a
dx2x using Parseval’s identity. (Dec 2008)
28) Find the Fourier transform of ( )xf if ( )
=,0
,1xf
if
if
0ax
ax
>><
. Hence
deduce that ∫∞
=
02
dt2
t
tsin π . (May/June 2009)
29) Find the Fourier Cosine transform of 22xae− for any a>0 and hence
prove that 2/xe2− is self-reciprocal under Fourier Cosine transform.
(May/June 2009)
30) Find the Fourier transform of ( )
−=
,0
,2x2axf if
if
0ax
ax
>><
. Hence
deduce that ∫∞
=−
04
dt3t
tcosttsin π. (May/June 2009)
31) Find
−axeCF ,
+ 2x1
1CF and
+ 2x1
xCF . (Hence CF stands for
Fourier Cosine transform) (May/June 2009)
UNIT – IIIPARTIAL DIFFERENTIAL EQUATIONS
9
TWO MARKS
1) Solve yx
zsin
2
2
=∂∂
. (May/June 2007)
2) Find the complete integral of pqp
y
q
x
pq
z ++= . (M/J 2007) (Dec 2008)
3) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the
equation ( ) ( ) ( )α2222 cotzbyax =−+− .(Nov/Dec 2007)(May/Jun 2009)
4) Find the complete solution of the PDE 0422 =−+ pqqp .(Nov/Dec2007)
5) Form the PDE of all spheres whose centers lie on the z-axis. (N/D 2006)
6) Find complete integral of the PDE ( ) ( ) zqypx −=−+− 321 .(N/D’06)
7) obtain the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the
equation ( ) ( ) 1222 =+−+− zbyax . (Nov/Dec 2003)
8) Find the general solution of 091242
22
2
2
=∂∂+
∂∂∂−
∂∂
y
z
yx
z
x
z.(Nov/Dec 2003)
9) Eliminate the arbitrary function ‘f ’ from
=
z
xyfz and form the PDE.
(Apr/May 2004)
10) Find the Complete integral of p+q=pq. (Apr/May 2004)
11) Find the PDE of all planes passing through the origin. (N/D 2005)
12) Find the particular integral of ( ) ( )yxZDDDDDD 2s i n1 243 3223 +=′+′−′− . (Nov/Dec 2005)
13) Find the PDE of all the planes having equal intercepts on the x & y axis.
(Nov/Dec 2005)
14) Find the solution of 222 zqypx =+ . (Nov/Dec 2005)
15) Find the PDE of all spheres having their centers on the line x=y=z .
(Oct/Nov 2002)
16) Solve 08423
3
2
3
2
3
3
3
=∂∂+
∂∂∂−
∂∂∂−
∂∂
y
z
yx
z
yx
z
x
z. (Oct/Nov 2002)
17) Solve ( ) 023 323 =′+′− ZDDDD . (Apr/May 2003)
10
18) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from
( )( )2222 byaxz ++= . (Nov/Dec 2004)
19) Solve ( ) 012 =−′+′− ZDDDD (Nov 2004)
20) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from nn byaxz += . (Apr/May 2005) (Dec 2008)
21) Solve ( ) 03 323 =′−′+ ZDDDD22) Form the PDE by eliminating the arbitrary function from z=f(xy).
23) Write down the complete solution of 221 qpcqypxz ++++= .
24) Form the partial differential equation by eliminating the arbitrary constants
‘a’ and ‘b’ from 3by3axz += (Apr 1995) (Dec 2008)
25) Find the singular solution of 12q2pqypxz ++++= . (Dec 2008)
26) Find the general solution of zqypx =+ . (Dec 2008)
27) Find the particular integral of y4x3eZDD42D +=
′− .(Dec 2008)
28) Solve 1qp =+ (Dec 2008)
29) Solve 0ZD2D3D =
′++ (Dec 2008)
30) Form the p.d.e by eliminating a and b from ( ) byxaz ++= .(Dec 2008)
31) Solve yxqp +=+ (Dec 2008)
32) Give the general solution of 0yx
z2=
∂∂∂ .(Dec 2008)
33) Solve 0Z2D2DD32D =
′+′+ (Dec 2008)
34) Form the partial differential equation by eliminating f from the relation
yx2y2xfz ++
+= .(May/June 2009)
SIX MARKS.
1) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the expression ( ) ( ) 2222 czbyax =+−+− . (May/June 2007)
11
2) Solve ( ) ( )yxZDDDD +=′+′− 2s i n5252 22. (May/June 2007)
3) Solve ( ) ( ) ( )2222 yxzqzxypzyx −=+++ . (May/June 2007)
4) Solve p(1+q)=qz. (May/June 2007)
5) Solve zxxyqpzyx 22)( 222 =+−− . (Nov/Dec 2007) (May/June 2009)
6) Solve ( ) yxeyxZDDDD 4322 322 +++=′−′− . (Nov/Dec 2007)
7) Solve ( ) 22222 yxqpz +=+ . (Nov/Dec 2007)
8) Solve yxzDDDD s i n)43( 22 +=′−′+ . (Nov/Dec 2007)
9) Find the singular integral of 22 qpqpqypxz ++++= . (Nov/Dec 2006)
10) Solve ( ) xyZDDDD s i n65 22 =′+′− . (Nov/Dec 2006)
11) Solve ( ) ( ) ( )xyqzxpyz 322443 −=−+− .(N/D 2006), (N/D 2003)
12) Solve yxeyxzDDDD −+=′+′+ 222 )2( . (Nov/Dec 2006)
13) Find the singular integral of PDE 22 qpqypxz −++= .(N/D 2003)
14) Solve ( )yxezDDDD yx 2s i n3)54( 222 −+=′−′+ −.(N/D 2003)
15) Find the general solution of ( ) ( ) ( )222222 xyzqzxypyzx −=−+− .
16) Solve 22322 )2()2332( yx eezDDDDDD −+=+′+−′+′− . (N/D 2003)
17) Solve ( ) ( ) ( )yxzqxzypzyx −=−+− . (Apr/May 2004)
12
18) Solve ( ) yxeyxzDDDD +++=′−′− 2323 2s i n)67( . (A/M 2004)
19) Solve ( ) ( ) 22 yxzqyxzpyx +=−++ . (Nov/Dec 2005)
20) Solve yxezDDDDDD +=+′++′+′+ 222 )1222( . (N/D 2005)
21) Solve 221 qpqypxz ++++= .
(May/June 2009) (Nov/Dec 2005),(Apr/May 2004)
22) Solve xyy
z
yx
z
x
zcos6
2
22
2
2
=∂∂−
∂∂∂+
∂∂
. (Nov/Dec 2005)
23) Solve ( ) ( ) ( )yxzqxzyxpyzyx +=−++++ 2222 .(N/D 2005)
24) Solve ( )yxezDDDD yx −+=′−′− + 4s i n)2 0( 522 .(Nov/Dec 2005)
25) Solve 22 qpz += . (Nov/Dec 2005)
26) Solve yxeyxzDDDD ++=′−′+ 3222 )6( . (Nov/Dec 2005)
27) Form the PDE by eliminating the arbitrary functions f and g in
( ) ( )yxgyxfz 22 33 −++= . (Oct/Nov 2002)
28) Solve ( ) ( ) ( ) ( )yxyxqxyzpxzy −+=−+− . (Oct/Nov 2002)
29) Solve yex yzDDDD x ++=′−′− 622 )3 0( .
30) Solve 222 1 qpz ++= . (April 1996) (Dec 2008)
31) Solve ( ) ( ) zxqyxpzy +=+−− 22 . (Apr/May 2003)
32) Form the PDE by eliminating the arbitrary functions f and g in
( ) ( )xgyyfxz 22 +=
33) Form the p.d.e by eliminating the function f and g from
( ) ( )y2xxgy2xfz +++= (Dec 2008)
34) Solve zxqyp =+ (Dec 2008)
35) Solve 2z2y2q2x2p =+ (Dec 2008)
36) Solve ( ) y2x3y2xsinzD2D23D ++=
′− (Dec 2008)
37) Form the p.d.e by eliminating the arbitrary function f and g from
( ) ( )y2xxgy2xfz +++= (Dec 2008)
13
38) Solve 0xzxyqp2z2y =+−
+ (Dec 2008)
39) Solve xcosyt6sr =−+ (Dec 2008)
40) Obtain complete solution of the equation pq2qypxz −+= (Dec 2008)
41) Solve ( )yx2c o sz2D6DD2D +=
′−′+ . (Dec 2008)
42) Solve xyyzqxzp =+ (Dec 2008)
43) Solve yx2ez2D2DD52D2 +=
′+′− (Dec 2008)
44) Find the complete solution of 2zpqxy = (May/June 2009)
45) Solve the equation ( )y2xs i nyxeZ2D2D +−=
′− . (May/June 2009)
UNIT-IVAPPLICATIONS OF PDE
TWO MARKS
1) Classify the following second order partial differential equations:
i) 01686442
22
2
2
=−∂∂−
∂∂−
∂∂+
∂∂∂+
∂∂
uy
u
x
u
y
u
yx
u
x
u. (Apr/May 2003)
ii) 22
2
2
2
2
∂∂+
∂∂=
∂∂+
∂∂
y
u
x
u
y
u
x
u . (Apr/May 2003)
iii) 0322 22 =−++− uuuxxyuuy xyyxyxx . (Nov/Dec 2003) iv) 07222 =++++ yxyyxx uuuuy . (Nov/Dec 2003) v) 0=+ yyxx xuu . (Apr/May 2004) 2) Classify the partial differential equations
t
u
x
u
∂∂=
∂∂
22
2 1
α. (May/June 2007)
3) Classify the following PDE i) xuxuux yyxyxx =+−+ 4)1( 2 . (March 1998), (Apr/May 2004) ii) 02)1(2 22 =−+++ xyyxyxx uuyxyuux . (Dec 1998)
14
4) What is the constant 2a in the wave equation xxtt uau 2= ? (N/D 2004)
5) In the diffusion equation 2
22
x
u
t
u
∂∂=
∂∂ α what does 2α stand for?
(N/D 2005) (Dec 2008)6) What is the basic difference between the solutions of one dimensional
wave equation and one dimensional heat equation? (Nov/Dec 2005)
7) What are the possible solutions of one dimensional wave equation?
(May/June 2006)
8) Explain the various variables involved in one dimensional wave equation.
(April 1995),(Nov 1995)
9) A tightly stretched string of length 2L is fastened at both ends. The
midpoint of the string is displaced to a distance ‘b’ and released from rest
in this position. Write the initial conditions. (May 2006)
10) Write the initial conditions of the wave equation if the string has an initial
displacement but no initial velocity. (A.U.Tri. Nov/Dec 2008)
11) Write the boundary conditions and initial conditions for solving the vibration
of string equation , if the string is subjected to initial displacement f(x) and
initial velocity g(x). (Nov/Dec 2006),(April 1998)
12) State one dimensional heat equation with initial and boundary conditions.
(Nov/Dec 2006)
13) In steady state conditions derive the solution of one dimensional heat flow
equation. (Nov/Dec 2005)
14) An insulated rod of length 60 cm has is ends A and B maintained at C20
and C80 respectively. Find the steady state solution of the rod.
(Nov/Dec 2003)
15) A rod 30 cm long has its ends A and B kept at C20 and C80
respectively until steady state conditions prevail. Find the steady state
temperature in the rod. (Apr/May 2004)
16) State any two laws which are assumed to derive one dimensional heat
equation. (Nov/Dec 2004)
17) State Fourier law of heat conduction. (Apr/May 2005)
18) What are the possible solutions of one dimensional heat equation?
(May 2000) (May/June 2009)
15
19) How many boundary conditions are required to solve completely
2
22
x
u
t
u
∂∂=
∂∂ α (April 1995)
20) Define temperature gradient. (Nov 1995)
21) State the assumptions made in the derivation of one dimensional wave
equation. (April 1995), (Nov 1995),(Nov 2007) (Dec 2008)
22) Write the steady state heat flow equation in two dimension in Cartesian &
Polar form. (Nov/Dec 2005)
23) Write any two solutions of the Laplace equation obtained by the method of
separation of variables. (April 2003)
24) In two dimensional heat flow, the temperature at any point is independent
of which coordinate?
25) Explain the term steady state.
26) Classify the p.d.e 0yyuxyu2x25xxu2x42x1 =+
++
+
+ (Dec 2008)
27) State the empirical laws used in deriving one-dimensional heat flow
equation. (Dec 2008)
28) Write the product solutions of 0urrurru2r =++ θ θ . (Dec 2008)
29) What is the equation governing the two dimensional heat flow steady state
and also write its solution. (Dec 2008)
30) Classify the p.d.e y3x2e2y
u2
yx
u22
2x
u2 +=∂
∂+∂∂
∂+∂
∂. (Dec 2008)
31) Write the various possible solutions of the Laplace equation in two
dimensions. (Dec 2008)
32) A infinitely long uniform plate is bounded by the edges lx,0x == and the
ends right angles to them. The breadth of the edges 0y = is l and is
maintained at ( )xf . All the other edges are kept at .C0 Write down the
boundary condition in mathematical form. (Dec 2008)
33) Write any two assumptions made while deriving the partial differential
equation of transverse vibrations of a string. (Dec 2008)
34) Define steady state. Write the one dimensional heat equation in steady
state. (Dec 2008)
16
35) Write all the solutions of Laplace equation in Cartesian form, using the
method of separation of variables. (Dec 2008)
36) Verify that ( ) ( )atcoshxcoshy λλ −= is a solution of .2
22
2
2
x
ya
t
y
∂∂=
∂∂
(M/J’09)
12 MARKS
1) A tightly stretched string of length ‘ l ’ has its ends fastened at x=0 & x=l .
The midpoint of the string is then taken to a height ‘ h ’ and then released
from rest in that position. Obtain an expression for the displacement of the
string at any subsequent time. (Nov 2002)
2) A tightly stretched flexible string has its ends fixed at x=0 and x=l. At time
t=0 , the string is given a shape defined by ),()( 2 xlkxxf −= where k is a
constant , and then released from rest. Find the displacement of any point
x of the string at any time t > 0. (April 2003)
3) A tightly stretched string with fixed end points x=0 and x=l is initially in a
position given by
=
l
xyxy
π30 sin)0,( . It is released from rest from this
position. Find the displacement at anytime ‘ t ’. (Nov 2004)
4) A tightly stretched string of length ‘ 2l ’ has its ends fastened at x=0 , x=2l.
The midpoint of the string is then taken to height ‘ b ’ and then released
from rest in that position. Find the lateral displacement of a point of the
string at time ‘ t ’ from yhe instant of release. (May 2005)
5) A string of length ‘ l ’ has its ends x=0 , x=l fixed. The point where 3
lx = is
drawn aside a small distance ‘ h ’,the displacement ),( txy satisfies
.2
22
2
2
x
ya
t
y
∂∂=
∂∂
Find ),( txy at any time ‘ t ’.
6) An elastic string of length ‘ 2l ’ fixed at both ends is disturbed from its
equilibrium position by imparting to each point an initial velocity of
magnitude ).2( 2xlxk − Find the displacement function ),( txy . (May ‘06)
7) A uniform string is stretched and fastened to two points ‘ l ’ apart. Motion
is started by displacing the string into the form of the curve ),( xlkxy −=
and then releasing it from this position at time t=0. Find the displacement
17
of the point of the string at a distance ‘ x ’ from one end at time ‘ t ’.
(A.U.Tri. Nov/Dec 2008) (Dec 2008) (May/June 2009)
8) If a string of length ‘ l ’ is initially at rest in its equilibrium position and each
of its points is given a velocity ‘ v ’ such that
−=
)( xlc
cxv
for
for
lxl
lx
<<
<<
2
20
show that the displacement at any time‘ t ’ is given by
+−= ...
3sin
3sin
3
1sinsin
4),(
33
2
l
at
l
x
l
at
l
x
a
cltxy
πππππ
. (Nov/Dec2008)
9) A string is stretched between two fixed points at a distance 2l apart and the
points of the string are given initial velocities ‘ v ’ where
−=
)2( xll
cl
cx
v in
in
lxl
lx
2
0
<<<<
‘ x ’ being the distance from one end point .Find the
displacement of the string at any subsequent time.
(April/May 2004)
10) The ends A and B of a rod ‘ l ’ cm long have the temperatures C40 and
C90 until steady state prevails. The temperature at A is suddenly raised
to C90 and at the same time that at B is lowered to C40 . Find the
temperature distribution in the rod at time ‘ t ’ . Also show that the
temperature at the midpoint of the rod remains unaltered for all time ,
regardless of the material of the rod. (April 2003)
11) A metal bar 10 cm long with insulated sides , has its ends A and B kept at
C20 and C40 until steady state conditions prevail. The temperature at A
is then suddenly raised to C50 and at the same instant that at B is
lowered to C10 . Find the subsequent temperature at any point of the bar
at any time . (Nov/Dec 2005)
18
12) The ends A and B of a rod ‘ l ’cm long have their temperatures kept at
C30 and C80 , until steady state conditions prevail. The temperature at
the end B is suddenly reduced to C60 and that of A is increased to C40
. Find the temperature distribution in the rod after time ‘ t ’. (M/J’ 07)
13) The boundary value problem governing the steady state temperature
distribution in a flat, thin , square plate is given by
,02
2
2
2
=∂∂+
∂∂
y
u
x
u ax <<0 , ay <<0
0)0,( =xu ,
=
a
xaxu
π3sin4),( , ax <<0
0),0( =yu , 0),( =yau , ay <<0 . Find the steady-state
temperature distribution in the plate. (Nov 2002)
14) A rectangular plate with insulated surface is 10 cm wide so long compared
to its width that it may be considered infinite length. If the temperature
along short edge y=0 is given by
=10
sin8)0,(x
xuπ
when 100 << x , while
the two long edges x=0 and x=10 as well as the other short edge are kept
at C0 , find the steady state temperature function ),( yxu .
(Nov 2003)
15) An infinitely long rectangular plate with insulated surface is 10 cm wide.
The two long edges and one short edge are kept at zero temperature while
the other short edge x=0 is kept at temperature given by
−=
)10(20
20
y
yu
for
for
105
50
≤≤≤≤
y
y . Find the steady state temperature in the plate.
(Nov/Dec 2005), (Nov 2004) (Dec 2008)
16) A rectangular plate with insulated surface is 10 cm wide and so long
compared to its width that it may be considered infinite in length without
introducing appreciable error. The temperature at short edge y=0 is given
19
by
−=
)10(20
20
x
xu
for
for
105
50
≤≤≤<
x
x and all the other three edges are
kept at C0 . Find the steady state temperature at any point in the plate.
(May 2005)
17) Find the steady state temperature distribution in a rectangular plate of
sides a and b insulated at the lateral surface and satisfying the boundary
conditions 0),(),0( == yauyu for by ≤≤0 , 0),( =bxu and )()0,( xaxxu −=
for ax ≤≤0 . (Nov/Dec 2005)
18) An infinitely long plate in the form of an area is enclosed between the lines
π== yy ,0 for positive values of x. The temperature is zero along the
edges π== yy ,0 and the edge at infinity. If the edge x=0 is kept at
temperature ‘ Ky(l-y)’ ’ find the steady state temperature distribution in the
plate. (May 2006)
19) An infinitely long uniform plate is bounded by two parallel edges and an
end at right angle to them. The breadth of this edge x=0 is π , this end is
maintained at temperature as )( 2yyKu −= π at all points while the other
edges are at zero temperature . Find the temperature ),( yxu at any point
of the plate in the steady state.
20) A rod of length ‘‘l ’ has its ends ‘A’ and ‘B’ kept at C0 and C120
respectively until steady state conditions prevail. If the temperature at ‘B’
is reduced to C0 and kept so while that of ‘A’ is maintained, find the
temperature distribution in the rod. (Dec 2008)
21) Find the steady state temperature in a circular plate of radius ‘a’ cm, which
has one half of its circumference at C0 and the other half at C100 .
(Dec 2008)
22) Find the steady state temperature distribution in a square plate bounded
by the lines 20y,20x,0y,0x ==== . Its surfaces are insulated, satisfying
the boundary conditions ( ) ( ) ( ) ( ) ( )x20x20,xu&00,xuy,20uy,0u −==== .
(Dec 2008)
23) A rectangular plate with insulated surface is 10 cm wide and so long
compared to its width that it may be considered infinite in length without
introducing appreciable error. If the temperature of the short edge y=0 is
20
given by xu = for 5x0 ≤≤ and ( )x10 − for 10x5 ≤≤ and the two long
edges x=0,x=10 as well as the other short edges are kept at C0 . Find the
temperature ( )y,xu at any point ( )y,x of the plate in the steady state.
(May/June 2009)
UNIT -VZ-TRANSFORM
TWO MARKS
1) Find
!n
aZ
n
in Z-transform. (Nov/Dec 2005)
2) Find [ ]iateZ − using Z-transform. (Nov/Dec 2005)3) State and prove initial value theorem in Z-transform.(M/J 2006)(Dec2008)
4) Find the Z-transform of (n+1)(n+2). (May/June 2006)
5) Find the Z-transform of (n+2). (Nov/Dec 2006)
6) State the final value theorem in Z-transform. (Nov/Dec 2006)
7) Find
n
Z1
. (May/June 2007)
8) Evaluate
++
−
10721
zz
zZ . (May/June 2007)
9) Prove that [ ]az
zaZ n
−= . (Apr/May 1999), (Apr/May 2000)
10) Prove that [ ]2
1
)( az
znaZ n
−=−
.
11) Prove that zek
Z1
!
1 =
.
12) Find [ ]cbnanZ ++2 .
13) Find the initial and final values of the function 2
1
25.01
1)( −
−
−+=
z
zzF .
14) Find the Z-transform of i) )()( onnfnf −= ii) )()( onnunf −= iii) )1()( 1 += + nuanf n iv) )()( 1 nunanf n−= .
15) What is the Z-transform of )(3
1nu
n
−
.
16) State the convolution property of Z-transform. (Dec 2008)
17) State and prove shifting theorem of Z-transform.
18) Find the Z-transform of 2
cos3πnn .
19) Find the Z-transform of ).0,( ≠baab n
20) Prove that [ ] [ ])0()()( fzfZTtfZ −=+ .
21
21) Find the Z-transform of 1n
1
+. (Dec 2008)
22) Prove that ( )( ) na1n
2az
2z1Z +=
−− (Dec 2008)
23) Find the difference equation from ( ) ( ) n2nBAny += (Dec 2008)24) State initial value theorem on Z-transform (Dec 2008) (May/June 2009)
25) Find
+−
92z
z1Z . (Dec 2008)
26) Define unit impulse sequence and find its Z-transform. (Dec 2008)27) Define convolution of two sequences. (Dec 2008)
28) Find the inverse Z-transform of 42z
2z
+ (Dec 2008)
29) From the difference equation of 1y,n2ny1ny 0 ==−+ , find ny in terms of z. (Dec 2008)
30) Find ( )( )nfZ ,where ( ) nnf = for n= 0, 1, 2, …. (May/June 2009)
SIX MARKS
1) Find
−−−
)2()1( 2
31
zz
zZ using partial fraction. (N/D2005)(Dec2008)
2) Solve the difference equation 0)(4)1(4)2( =++−+ kykyky where .0)1(,1)0( == yy (Nov/Dec 2005)
3) Prove that
−=
+ 1log
1
1
z
zz
nZ . (Nov/Dec 2005)
4) State and prove second shifting theorem in Z-transform. (Nov/Dec 2005)
5) Using convolution theorem evaluate inverse Z-transform of
−− )3)(1(
2
zz
zZ . (Dec 2008) (May/June 2006)
6) Using Z-transform solve 2,0)2(4)1(3)( ≥=−−−+ nnynyny given that .2)1(,3)0( −== yy (May/June 2006)
7) Find
−++−−
2
21
)1)(1(
)2(
zz
zzzZ by using method of partial fraction.(N/D 2006)
8) Find Z-transform of )2)(1(
1
++ nn . (Nov/Dec 2006)
9) Using convolution theorem evaluate
−−
−)2)(1(
21
zz
zZ .(Nov/Dec 2006)
10) Find Z-transform of na and θnan cos . (May/June 2007)11) Using the Z-transform method solve 22 =++ nn yy given that 010 == yy .
(May/June 2007)12) State and prove final value theorem in Z-transform.(May/Jun 2007)
13) Find the inverse Z-transform of 3)1(
)1(
−+
z
zz. (May/June 2007)
14) State and prove first shifting theorem on Z-transform. Also find [ ]teZ at− .
22
15) Use Z-transform to solve nnnn yyy 2127 12 =+− ++ given 010 == yy .
(Dec 2008)16) Find [ ]iateZ − and hence deduce the values of [ ]atZ cos and [ ]atZ sin .
17) Find
−+−
)1()1( 21
zz
zZ .
18) Prove that [ ] [ ]1−−= pp nZdz
dznZ where p is any positive integer. Deduce
that [ ]2)1( −
=z
znZ and [ ]
3
22
)1(
2
−+=
z
znZ .
19) Find the inverse Z-transform of )4)(2(
32 2
−++zz
zz.
20) Find the inverse Z-transform of ( )( )2n1n ++ . (Dec 2008)21) Using Z-transforms, solve ( ) ( ) ( ) ,1n,0ny41ny32ny ≥=−+++ given that
( ) 30y = and ( ) 21y −= . (Dec 2008)
22) Find the Z-transform of the sequence 1n
1nf +
= (Dec 2008)
23) Find the inverse Z-transform of ( )( )22z
z42z2ZF
−
+= using residue theorem.
(Dec 2008)
24) By using convolution theorem, prove the inverse of ( )( )bzaz
2z
++ is
( ) { }1na1nbab
n1 +−+−
− . (May/June 2009).
25) By the method of Z – transform solve ( ) ( ) ( ) n2ny91ny62ny =++++ given that ( ) 00y = and ( ) 01y = . (May/June 2009)
26) Find the Z – transform of θncos and hence find ( )θncosnZ . (May/June 2009)27) Solve the equation (using Z – transform) ( ) ( ) ( ) 36ny61ny52ny =++−+
given that ( ) ( ) .01y0y == (May/June 2009)
23