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Gujarat Technological UniversityWinter 2023 Examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Winter 2023 Paper Solution & PDF

B.E. · Common Engineering · Semester 2 · Subject Code: 3130005
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Total Marks70 MarksExternal theory exam
Passing Marks23 Marks33% minimum cutoff
Exam Duration2.5 Hours10:30 AM – 1:00 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Express (2+i3−i)2\left(\frac{2+i}{3-i}\right)^2 into Polar form.
3 Marks
(b)
Find and plot the fourth roots of (−1)(-1).
4 Marks
(c)
Solve (D2+DD′+D′−1)z=sin⁡(x+2y)(D^2 + DD' + D' - 1)z = \sin(x + 2y).
7 Marks

Question 2

14 MarksMedium
(a)
Determine aa and bb such that u=ax3+bxyu = ax^3 + bxy is harmonic.
3 Marks
(b)

Discuss the continuity of f(z)f(z) at the origin: f(z)={zˉz,if z≠00,if z=0f(z) = \begin{cases} \frac{\bar{z}}{z}, & \text{if } z \ne 0 \\ 0, & \text{if } z = 0 \end{cases}

4 Marks
(c)
Show that the function u=excos⁡yu = e^x \cos y is harmonic. Find the conjugate function vv and express u+ivu + iv as an analytic function of zz.
7 Marks
OR OPTION
(c)
Find the bilinear transformation which maps the points 1,−1,∞1, -1, \infty onto the points 1+i,1−i,11 + i, 1 - i, 1 respectively. Also, find its fixed points.
7 Marks

Question 3

14 MarksMedium
(a)
Find the radius of convergence of the power series ∑n=1∞zn2n+1\sum_{n=1}^\infty \frac{z^n}{2^n + 1}
3 Marks
(b)
Separate (i)i(\sqrt{i})^{\sqrt{i}} into real and imaginary parts.
4 Marks
(c)
State Cauchy’s Integral Theorem and use it to find ∫Ce2zz2+1 dz,where C is ∣z∣=12\int_C \frac{e^{2z}}{z^2 + 1} \, dz, \quad \text{where } C \text{ is } |z| = \frac{1}{2}
7 Marks
OR OPTION
(a)
Evaluate ∫02+iz2 dz\int_0^{2+i} z^2 \, dz along the line y=x2y = \frac{x}{2}.
3 Marks
(b)
Expand f(z)=cos⁡zf(z) = \cos z as a Taylor series about z=0z = 0.
4 Marks
(c)
Write Cauchy’s Integral formula and hence evaluate:

∫Csin⁡πz2+cos⁡πz2(z−1)(z−2) dz,where C is ∣z∣=3\int_C \frac{\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)} \, dz, \quad \text{where } C \text{ is } |z| = 3

7 Marks

Question 4

14 MarksMedium
(a)
Classify the singular point z=0z = 0 for function f(z)=1z4−4z2f(z) = \frac{1}{z^4 - 4z^2}.
3 Marks
(b)
Find the complete integral of p−3x2=q2−yp - 3x^2 = q^2 - y.
4 Marks
(c)

Obtain the Laurent’s series for the function f(z)=1z(1−z)f(z) = \frac{1}{z(1-z)} in the regions:

i∣z+1∣<1|z + 1| < 1
ii1<∣z+1∣<21 < |z + 1| < 2
iii∣z+1∣>2|z + 1| > 2
7 Marks
OR OPTION
(a)
Derive partial differential equation by eliminating aa and bb from z=(x−a)2+(y−b)2z = (x - a)^2 + (y - b)^2.
3 Marks
(b)
Using Cauchy’s residue theorem, evaluate ∫C5z−2z(z−1) dz,∣z∣=2\int_C \frac{5z - 2}{z(z - 1)} \, dz, \quad |z| = 2
4 Marks
(c)
Solve x(y−z)p+y(z−x)q=z(x−y)x(y - z)p + y(z - x)q = z(x - y).
7 Marks

Question 5

14 MarksMedium
(a)
Evaluate ∫0∞11+x2 dx\int_0^\infty \frac{1}{1 + x^2} \, dx using contour integration.
3 Marks
(b)
Solve (D2−4DD′+4D′2)z=e2x+3y(D^2 - 4DD' + 4D'^2)z = e^{2x + 3y}.
4 Marks
(c)
Find the solution of the wave equation ∂2y∂t2=c2∂2y∂x2\frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} such that y=acos⁡pty = a \cos pt when x=lx = l, and y=0y = 0 when x=0x = 0.
7 Marks
OR OPTION
(a)
Solve (2D2+5DD′+2D′2)z=0(2D^2 + 5DD' + 2D'^2)z = 0.
3 Marks
(b)
Solve p(1+q)=qzp(1 + q) = qz.
4 Marks
(c)
Using the method of separation of variable, find the solution of ∂u∂x=2∂u∂t+u\frac{\partial u}{\partial x} = 2\frac{\partial u}{\partial t} + u, u(x,0)=6e−3xu(x, 0) = 6e^{-3x}.
7 Marks
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About this Examination Paper & Attribution

Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Complex Variables and Partial Differential Equations (CVPD) (Winter 2023, B.E. · Common Engineering, Sem 2). Features complete 70-mark regular & remedial examination pattern, official marking distribution across all 5 questions, and direct 1-click official PDF download.

Transcribed for student exam preparation from Gujarat Technological University official examination archives. Questions, syllabus guidelines, and curriculum marking schemes remain the intellectual property of Gujarat Technological University.

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