UnFoldPrep
Browse Archive
Gujarat Technological UniversityWinter 2025 Examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Winter 2025 Paper Solution & PDF

B.E. · Common Engineering · Semester 2 · Subject Code: 3130005
Download Official GTU PDF
Share:
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)
Show that ∣z∣z∣−1∣≤∣arg⁡(z)∣|\frac{z}{|z|} - 1| \le |\arg(z)|.
3 Marks
(b)
Prove that u=x2−y2u = x^2 - y^2 and v=yx2+y2v = \frac{y}{x^2+y^2} are harmonic functions of (x,y)(x, y) but are not harmonic conjugates.
4 Marks
(c)
Find the bilinear transformation which maps the points z=1,i,−1z = 1, i, -1 into w=i,0,−iw = i, 0, -i respectively. Find invariant points and image of ∣z∣<1|z| < 1 under this mapping.
7 Marks

Question 2

14 MarksMedium
(a)
Evaluate the integral ∫Czˉ dz\int_C \bar{z} \, dz, where CC is right half of the circle z=2eiθ(−π2<θ<π2)z = 2e^{i\theta} \left(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\right).
3 Marks
(b)
Evaluate ∫03+iz2 dz\int_0^{3+i} z^2 \, dz along the parabola x=3y2x = 3y^2.
4 Marks
(c)
Explain inversion transformation. Find the image of 14≤y≤12\frac{1}{4} \le y \le \frac{1}{2} under mapping w=1/zw = 1/z. Show this translation on graph.
7 Marks
OR OPTION
(c)
State Cauchy integral theorem and Cauchy integral formula. Evaluate ∮csin⁡πz2+cos⁡πz2(z−1)(z−2) dz\oint_c \frac{\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)} \, dz, where c:∣z∣=3c : |z| = 3.
7 Marks

Question 3

14 MarksMedium
(a)
Determine the poles of the function and the residue at each pole: f(z)=z2(z−2)2(z−1)f(z) = \frac{z^2}{(z-2)^2(z-1)}.
3 Marks
(b)
Find the radii of convergence and region of convergence of the following:
i∑n=1∞zn2n+1\sum_{n=1}^\infty \frac{z^n}{2^{n+1}}
ii∑n=1∞n!nnzn\sum_{n=1}^\infty \frac{n!}{n^n} z^n
4 Marks
(c)

Find the Laurent series of 1z(z2−3z+2)\frac{1}{z(z^2 - 3z + 2)} for region:

i0<∣z∣<10 < |z| < 1
ii1<∣z∣<21 < |z| < 2
iii∣z∣>2|z| > 2
7 Marks
OR OPTION
(a)
Define: Singular point, Isolated singular point, Residue. Explain types of isolated singular points.
3 Marks
(b)
Expand f(z)=z−1z+1f(z) = \frac{z-1}{z+1} as Taylor series about the following points: a) z=0z = 0, b) z=1z = 1.
4 Marks
(c)
Evaluate following real integration using residue theorem: ∫−∞∞cos⁡xx2+1 dx\int_{-\infty}^\infty \frac{\cos x}{x^2 + 1} \, dx.
7 Marks

Question 4

14 MarksMedium
(a)
Eliminate the arbitrary function from the equation z=xy+f(x2+y2)z = xy + f(x^2 + y^2).
3 Marks
(b)
Solve y2p−xyq=x(z−2y)y^2 p - xyq = x(z - 2y).
4 Marks
(c)
Solve following non-linear partial differential equations using Charpit's method: px+qy=pqpx + qy = pq.
7 Marks
OR OPTION
(a)
Solve: ∂2z∂x2+z=0\frac{\partial^2 z}{\partial x^2} + z = 0 given that z=eyz = e^y and ∂z∂x=1\frac{\partial z}{\partial x} = 1 when x=0x = 0.
3 Marks
(b)
Solve following non-linear partial differential equations:
ap(1+q)=qzp(1 + q) = qz
bp2−q2=x−yp^2 - q^2 = x - y
4 Marks
(c)
Solve the following Partial differential Equation by Lagrange's Method: (x2−yz)p+(y2−zx)q=(z2−xy)(x^2 - yz)p + (y^2 - zx)q = (z^2 - xy).
7 Marks

Question 5

14 MarksMedium
(a)
Find the Particular integral of 4r+12s+9t=e3x−2y4r + 12s + 9t = e^{3x - 2y}.
3 Marks
(b)
Using method of separation of variables solve ∂u∂x=2∂u∂t+u\frac{\partial u}{\partial x} = 2 \frac{\partial u}{\partial t} + u given that u(x,0)=6e−3xu(x, 0) = 6e^{-3x}.
4 Marks
(c)
Show that u=sin⁡9tsin⁡(x/4)u = \sin 9t \sin(x/4) is a solution of a one dimensional wave equation.
7 Marks
OR OPTION
(a)
Classify following second order homogeneous partial differential equations as elliptic, parabolic or hyperbolic:
a∂2u∂t2+4∂2u∂x∂t+4∂2u∂x2=0\frac{\partial^2 u}{\partial t^2} + 4 \frac{\partial^2 u}{\partial x \partial t} + 4 \frac{\partial^2 u}{\partial x^2} = 0
b2∂2u∂t2+4∂2u∂x∂t+3∂2u∂x2=02 \frac{\partial^2 u}{\partial t^2} + 4 \frac{\partial^2 u}{\partial x \partial t} + 3 \frac{\partial^2 u}{\partial x^2} = 0
3 Marks
(b)
Solve ∂2z∂x2+3∂2z∂x∂y+2∂2z∂y2=x+y\frac{\partial^2 z}{\partial x^2} + 3 \frac{\partial^2 z}{\partial x \partial y} + 2 \frac{\partial^2 z}{\partial y^2} = x + y.
4 Marks
(c)
Derive the solutions of one dimensional wave equation.
7 Marks
College Exam Groups

Studying for Complex Variables and Partial Differential Equations?

Circulate this solved paper with KaTeX formulas and 1-click AI step solvers to your batchmates on WhatsApp or Telegram.

Related papers

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 2025, 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.

Download PDF