1
Seat No.: Enrolment
GUJARAT TECHNOLOGICAL UNIVERSITY
PD -SEMESTER 1 (NEW SYLLABUS) EXAMINATION- WINTER 2018
Subject Code: 2910002 Date: 03-01-2019
Subject Name: Complex variables and Partial Differential Equations
Time: 10:30 am to 01:30 pm Total Marks: 70
Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 Determine d such that 2 2 2 2 f x axy cx dxy y is
analytic.
04
Prove that 2 2 x y is harmonic function. 03
Find the image of the following curves under the mapping; the line
y x z 5.
07
Q.2 Find the bilinear transformation which maps the points onto the
points ,0. wi i .Hence find the image of 1. z Also find the invariants points
of this transformation.
07
Show that 0.
C
z dz where C is the boundary of the square whose vertices
are at the points 1 z z z i and
07
Q.3
Evaluate

2 2 sin cos

1 2 C
z z
dz
z z

where C is the circle 3. z
04
Evaluate


3 2
z
C
e
dz
z z where C is the circle
1
1 .
2
z
03

Expand the function 2
1

4 3
f z
z z


for z 3. in Laurrent's series.
07
Q.4
Evaluate
2
12 7

C 1 2 3
z
dz
z z

where C is the circle z 2.
07

Evaluate
2
0 5 3cos
d
by contour integration.
07
Q.5 Form the partial differential equation by eliminating the arbitrary constants
of 2 2z ax .
03
Solve
2
4 sin(3
u
x xy
y x



04
Solve 2 p 3q 04
Solve 2 2 p q 2. 03
Q.6 Solve 2 z pqxy by Charpit`s method. 07

Solve
2 2 2
2 2
2 2 2
z z z
x xy y
x x y y



07
2
Q.7
Solve
uu
xy



where 3 8 y u y .
07

Solve the equation 2 1
sin
sin
v v
r
r r





with the boundary
conditions 0 cos
v v
r a u r
r r




where 0u is a constant.
07