Download GTU B.Tech 2020 Summer 4th Sem 3140610 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 4th Sem 3140610 Complex Variables And Partial Differential Equations Previous Question Paper

Seat No.: ________
Enrolment No.___________


GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER? IV EXAMINATION ? SUMMER 2020
Subject Code: 3140610 Date:02/11/2020
Subject Name: Complex Variables and Partial Differential Equations
Time: 10:30 AM TO 01:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.



Mark
s
Q.1 (a) Show that the function
2
2
u x y is harmonic and find the
03
corresponding analytic function.

(b) Find the fourth roots of 1.
04

(c) (i) Find the image of the infinite strip 0 x 1 under the transformation
03
w iz .
1 Sketch the region.


1
04
(ii) Write the function f (z) z in the form f (z) u(r, ) iv(r, ).
z


Q.2 (a) Evaluate x2 ixy)dz from (1,1) to (2,4) along the curve x t,
2
y t .
03
C

(b) Find the bilinear transformation which transforms z
0
,
1
,
2
into
04
w ,
1 ,
0 i

(c)
dz
03
(i) Evaluate
, where C is z i 1, counter clockwise.
z 2 1
C


(n 2
!)
04
(ii) Find the radius of convergence of
n
z .
n1 (2n)!


OR


(c) (i) Find the roots of the equation 2
z 2iz (2 4i) .
0
03


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



Q.3 (a)
3
6
03
Find
, where C : z 2.
2
dz
z i
(z i)
C

(b)
1
04
Find the residues of f (z)
, has a pole at z = 3 and a pole
(z )
1 2 (z )
3
of order 2 at z = 1.

(c)
1
07
Expand f (z)
in a Laurent series for the regions (i) z 1
(z )(
1 z )
3
, (ii 1
) z 3 , (iii) z .
3


OR

Q.3 (a)
z
e
03
Evaluate
dz, where C : z 1 1.
z i
C

(b)
5z 2
04
Evaluate by using Cauchy's residue theorem
dz ; z .
2
z(z )
1
C
1

---

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


(ii 1
) z ,
2 (iii) z .
2



Q.4 (a) Solve
x
-
yq
p z
=
.
03

(b) Derive partial differential equation by eliminating the arbitrary constants
04
a and b from z ( 2
x a)( 2
y ).
b

(c) (i) Solve the p.d.e. r 3as 2 2
a t .
0
03


(ii) Find the complete integral of p 1
( q) qz.
04

OR

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

(b) Form the partial differential equation by eliminating the arbitrary function
04
from (x y z, 2
2
2
x y z ) .
0

(c) (i) Solve the p.d.e. s p q z .
xy
03


(ii) Solve by Charpit's method q 3 2
p .
04



Q.5 (a) Solve
2
2
2x3y
(D 2DD'D' )z e
03

(b) Solve the p.d.e.
3
y
u u
4 ,u( ,
0 y) e
8
using the method of separation
04
x
y
of variables.

(c) Find the solution of the wave equation u
c2
u , 0 x L with the
07
tt
xx
x
conditions u( ,
0 t) u(L,t) ;
0 t ,
0 u(x )
0
,
,u (x )
0
, 0
;
0 x .
L
L
t


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

2

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