Gujarat Technological UniversitySummer 2026 examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Summer 2026 Question Paper & PDF

B.E. · Common Engineering · Semester 3 · Subject code 3130005
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Total
70 marks
To pass
23 marks (33%)
Duration
2.5 Hours 2:30 PM – 5:00 PM
Questions
5 with internal choice
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3 of 25 questions on this paper were also asked in other years. 18 more are types of question GTU sets again with new data. Each is marked below with the years it appeared.See all 37 repeated 3130005 questions, most asked first

Q.1

14 Marks
(a)
3 marksSame type asked in 4 of 7 papers
Express the complex number (1+3i)2(1 + \sqrt{3}i)^2 in polar form.

Also asked in Summer 2024 Q1a, Winter 2023 Q1a, Summer 2023 Q3b OR

(b)
4 marksSame type asked in 5 of 7 papers
Solve the partial differential equation: (D2−DD′−12D′2)z=e2x−3y(D^2 - DD' - 12D'^2)z = e^{2x - 3y}.

Also asked in Winter 2025 Q5a, Summer 2024 Q5b OR, Winter 2023 Q5b, Summer 2023 Q5a

(c)
7 marksAsked in 3 of 7 papers
Show that u(x,y)=2x−x3+3xy2u(x, y) = 2x - x^3 + 3xy^2 is harmonic and find its harmonic conjugate v(x,y)v(x, y).

Also asked in Summer 2025 Q2c OR, Summer 2023 Q1c

Q.2

14 Marks
(a)
3 marksAsked in 3 of 7 papers
Discuss about the types of singularities of f(z)f(z).

Also asked in Winter 2025 Q3a OR, Summer 2024 Q3a

(b)
4 marksSame type asked in 4 of 7 papers
Evaluate ∫C(x2+ixy) dz\int_C (x^2 + ixy) \, dz; from A(1,1)A(1, 1) to B(2,4)B(2, 4) along the curve y=x2y = x^2.

Also asked in Winter 2025 Q2b, Winter 2024 Q2b, Winter 2023 Q3a OR

(c)
7 marksSame type asked in 7 of 7 papers
Find the bilinear transformation mapping z1=1,z2=0,z3=−1z_1 = 1, z_2 = 0, z_3 = -1 to w1=i,w2=∞,w3=1w_1 = i, w_2 = \infty, w_3 = 1.

Also asked in Winter 2025 Q1c, Summer 2025 Q2c, Winter 2024 Q1c, Summer 2024 Q1c, Winter 2023 Q2c OR, Summer 2023 Q2c OR

(c)
7 marksSame type asked in 3 of 7 papers
Determine an analytic function with real part u(x,y)=yx2+y2u(x, y) = \frac{y}{x^2 + y^2}.

Also asked in Summer 2025 Q1c, Summer 2023 Q1a

Q.3

14 Marks
(a)
3 marksSame type asked in 3 of 7 papers
Find all values of (−8i)1/3(-8i)^{1/3} using De Moivre's theorem.

Also asked in Winter 2024 Q1b, Winter 2023 Q1b

(b)
4 marksSame type asked in 4 of 7 papers
Evaluate ∫Ccos⁡πz2(z−1)(z−2) dz\int_C \frac{\cos \pi z^2}{(z - 1)(z - 2)} \, dz, where CC is ∣z∣=5/2|z| = 5/2.

Also asked in Winter 2025 Q2c OR, Summer 2024 Q2c OR, Winter 2023 Q3c OR

(c)
7 marksSame type asked in 7 of 7 papers

Find the Laurent series of 1z(z−1)(z−2)\frac{1}{z(z - 1)(z - 2)} for the regions:

i0<∣z∣<10 < |z| < 1, (ii) 1<∣z∣<21 < |z| < 2, (iii) ∣z∣>2|z| > 2.

Also asked in Winter 2025 Q3c, Summer 2025 Q3b OR, Winter 2024 Q2c, Summer 2024 Q3b, Winter 2023 Q4c, Summer 2023 Q2b, Summer 2023 Q3a OR

(a)
3 marks
State Cauchy-Riemann Equations and show that the function f(z)=xy+iyf(z) = xy + iy is nowhere analytic.
(b)
4 marksSame type asked in 3 of 7 papers
Find Maclaurin series expansion of f(z)=1z+3if(z) = \frac{1}{z + 3i} and its radius of convergence.

Also asked in Winter 2025 Q3b OR, Winter 2023 Q3b OR

(c)
7 marksSame type asked in 6 of 7 papers
Find the residues at singular points of f(z)=z2(z−1)(z+2)f(z) = \frac{z^2}{(z - 1)(z + 2)}. Using Cauchy's Residue theorem evaluate ∫Cf(z) dz\int_C f(z) \, dz, where CC is ∣z+1∣=1|z + 1| = 1.

Also asked in Winter 2025 Q3a, Summer 2025 Q3b, Summer 2025 Q4a, Winter 2024 Q3a OR, Summer 2024 Q3b OR, Summer 2023 Q3c

Q.4

14 Marks
(a)
3 marksSame type asked in 6 of 7 papers
Form partial differential equation from the equation z=xy+f(x2+y2)z = xy + f(x^2 + y^2) by eliminating arbitrary function.

Also asked in Winter 2025 Q4a, Summer 2025 Q5a, Winter 2024 Q4a OR, Summer 2024 Q4a, Summer 2023 Q4b OR

(b)
4 marksSame type asked in 3 of 7 papers
Classify the second order homogeneous partial differential equation ∂2z∂x2−∂2z∂x∂y−12∂2z∂y2=0\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial x \partial y} - 12\frac{\partial^2 z}{\partial y^2} = 0 as elliptic, parabolic or hyperbolic.

Also asked in Winter 2025 Q5a OR, Summer 2024 Q5a

(c)
7 marks
iSolve the linear partial differential equation xp+yq=3zxp + yq = 3z.
iiFind the complete integral of p2=q+xp^2 = q + x.
(a)
3 marksSame type asked in 4 of 7 papers
Form partial differential equation from z=ax+by+az = ax + by + a by eliminating the arbitrary constant aa and bb.

Also asked in Winter 2024 Q4a, Winter 2023 Q4a OR, Summer 2023 Q4b

(b)
4 marksSame type asked in 3 of 7 papers
Solve the second order homogeneous partial differential equation: zxx−2zxy−8zyy=0z_{xx} - 2z_{xy} - 8z_{yy} = 0.

Also asked in Winter 2024 Q5b OR, Winter 2023 Q5a OR

(c)
7 marksSame type asked in 4 of 7 papers
Using the Charpit's method solve the non linear partial differential equation: px+qy=pqpx + qy = pq.

Also asked in Winter 2025 Q4c, Winter 2024 Q4c, Summer 2024 Q4c

Q.5

14 Marks
(a)
3 marksSame type asked in 2 of 7 papers
Solve the partial differential equation using the method of separation of variables: ux+2uy=0u_x + 2u_y = 0, given that u(x,0)=4e−xu(x, 0) = 4e^{-x}.

Also asked in Summer 2025 Q5c

(b)
4 marksSame type asked in 6 of 7 papers
Evaluate ∫Ce−zz+1 dz\int_C \frac{e^{-z}}{z + 1} \, dz, where CC is (i) ∣z−1∣=1|z - 1| = 1, (ii) ∣z+1∣=1|z + 1| = 1.

Also asked in Summer 2025 Q3a, Winter 2024 Q3c, Winter 2024 Q3b OR, Summer 2024 Q3c, Winter 2023 Q3c, Winter 2023 Q4b OR, Summer 2023 Q2c, Summer 2023 Q3c OR

(c)
7 marks
A tightly stretched string with fixed ends x=0x = 0 and x=Lx = L is initially in a position given by u(x,0)=u0sin⁡3(πx/L)u(x, 0) = u_0 \sin^3(\pi x / L). If it is released at rest from this position, find the displacement u(x,t)u(x, t).
(a)
3 marksAsked in 4 of 7 papers
Using the method of separation of variables, solve ux=2ut+uu_x = 2u_t + u with u(x,0)=6e−3xu(x, 0) = 6e^{-3x}.

Also asked in Winter 2025 Q5b, Winter 2024 Q5c OR, Winter 2023 Q5c OR

(b)
4 marks
Find Upper bound of absolute value of the integral ∮Cez dz\oint_C e^z \, dz, where CC is a line segment joining point (0,0)(0, 0) and (1,22)(1, 2\sqrt{2}).
(c)
7 marksSame type asked in 3 of 7 papers
Solve Laplace's equation in square lamina with u(0,y)=u(π,y)=u(x,π)=0u(0, y) = u(\pi, y) = u(x, \pi) = 0 and u(x,0)=T0u(x, 0) = T_0, constant.

Also asked in Summer 2025 Q5c OR, Summer 2024 Q5c

More Complex Variables and Partial Differential Equations papers

Questions repeat across sessions. Working through two or three papers shows you which topics GTU asks most.

Winter 202570 marks · 5 questions
Summer 202570 marks · 5 questions
Winter 202470 marks · 5 questions
Summer 202470 marks · 5 questions

Other Semester 3 subjects

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2130002
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2130003
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BE03000051
Professional Communication and Ethics
BE03000061
Indian Constitution

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About this paper

Official Gujarat Technological University (GTU) examination paper for Complex Variables and Partial Differential Equations (CVPD) (Summer 2026, B.E. · Common Engineering, Sem 3). 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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