Download GTU B.Tech 2020 Summer 3rd Sem 3130005 Complex Variables And Partial Differential Equations Question Paper

Download GTU (Gujarat Technological University Ahmedabad) B.Tech/BE (Bachelor of Technology/ Bachelor of Engineering) 2020 Summer 3rd Sem 3130005 Complex Variables And Partial Differential Equations Previous Question Paper

Seat No.: ________
Enrolment No.___________


GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER? III EXAMINATION ? SUMMER 2020
Subject Code: 3130005 Date:27/10/2020
Subject Name: Complex Variables and Partial Differential Equations
Time: 02:30 PM TO 05:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.



Marks
Q.1 (a) If u x3 3xy is find the corresponding analytic function f (z) u .
iv
03

(b) Find the roots of the equation 2
z 5
( i)z 8 i .
0
04

(c) (i) Determine and sketch the image of z 1 under the transformation
03
w z .i


(ii) Find the real and imaginary parts of f (z)
2
z 3 .
z
04


Q.2 (a) Evaluate x2 iy2)dz along the parabola
2
y 2x from (1,2) to (2,8).
03
C

(b) Find the bilinear transformation that maps the points z ,i,0 into
04
w ,
0 i, .

(c)
z
e dz
03
(i) Evaluate
, where C is the circle z 1/ 2.
z 1
C


n2
1
04
(ii) Find the radius of convergence of
n
1 z .
n1
n


OR


(c) (i) Find the fourth roots of 1.
03


04
(ii) Find the roots of log z i .
2



Q.3 (a)
1
03
Find dz , where C : z .1
z 2
C

(b)
1
04
For f (z)
, find Residue of f(z) at z=1.
(z )
1 2 (z )
3

(c)
1
07
Expand f (z)
in a Laurent series for the regions (i) z 2 ,
(z )(
2 z )
4
(ii)2 z 4 , (iii) z .
4


OR

Q.3 (a)
z 4
03
Find
dz , where C : z 1 1.
z 2 2z 5
C

(b)
2z
e
04
Evaluate using Cauchy residue theorem
dz; C: 4 2
x 9 2
y
.
16
(z 3
)
1
C

(c)
1
07
Expand f (z)
in Laurent's series for the regions
z(z )(
1 z )
2
(i) z ,
1 (ii 1
) z ,
2 (iii) z .
2
1

---

Q.4 (a) Solve xp yq x - y.
03



(b) Derive partial differential equation by eliminating the arbitrary constants
04
a and b from z ax by
.
ab

(c) (i) Solve the p.d.e. 2r 5s 2t .
0
03


(ii) Find the complete integral of 2
p
.
qz
04

OR

Q.4 (a) Find the solution of 2
2
2
x p y q z .
03

(b) Form the partial differential equation by eliminating the arbitrary function
04
y
from z .
x

(c) (i) Solve the p.d.e. 2
D '2
D D D 'z .
0
03


(ii) Solve by Charpit's method
2
yzp q .
0
04



Q.5 (a) Solve (2 2
D 5
'
DD 2
'2
D )z
(
24 y x).
03

(b) Solve the p.d.e. u u (
2 x y u
) using the method of separation of
04
x
y
variables.

(c) Find the solution of the wave equation u
c2
u , 0 x with the
07
tt
xx
initial
and
boundary
conditions
u( ,
0 t) u( ,t) ;
0 t ,
0
u(x )
0
, k(sin x sin 2 ),
x u (x )
0
, 0
;
0 x . ( 2
c )
1
t


OR
Q.5 (a) Solve the p.d.e. r s q z .
0
03
(b) Solve u
2
u u given
3
x
u(x )
0
, e
4
using the method of separation
04
x
t
of variables.
(c) Find the solution of u
c2
u together with the initial and boundary
07
t
xx
conditions u( ,
0 t) u( ,
2 t) ;
0 t 0 and u(x )
0
,
0
;
10 x .
2

2

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