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Gujarat Technological UniversitySummer 2026 Examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Summer 2026 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 Hours2:30 PM – 5:00 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Express the complex number (1+3i)2(1 + \sqrt{3}i)^2 in polar form.
3 Marks
(b)
Solve the partial differential equation: (D2−DD′−12D′2)z=e2x−3y(D^2 - DD' - 12D'^2)z = e^{2x - 3y}.
4 Marks
(c)
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).
7 Marks

Question 2

14 MarksMedium
(a)
Discuss about the types of singularities of f(z)f(z).
3 Marks
(b)
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.
4 Marks
(c)
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.
7 Marks
OR OPTION
(c)
Determine an analytic function with real part u(x,y)=yx2+y2u(x, y) = \frac{y}{x^2 + y^2}.
7 Marks

Question 3

14 MarksMedium
(a)
Find all values of (−8i)1/3(-8i)^{1/3} using De Moivre's theorem.
3 Marks
(b)
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.
4 Marks
(c)

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

Question 4

14 MarksMedium
(a)
Form partial differential equation from the equation z=xy+f(x2+y2)z = xy + f(x^2 + y^2) by eliminating arbitrary function.
3 Marks
(b)
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.
4 Marks
(c)
iSolve the linear partial differential equation xp+yq=3zxp + yq = 3z.
iiFind the complete integral of p2=q+xp^2 = q + x.
7 Marks
OR OPTION
(a)
Form partial differential equation from z=ax+by+az = ax + by + a by eliminating the arbitrary constant aa and bb.
3 Marks
(b)
Solve the second order homogeneous partial differential equation: zxx−2zxy−8zyy=0z_{xx} - 2z_{xy} - 8z_{yy} = 0.
4 Marks
(c)
Using the Charpit's method solve the non linear partial differential equation: px+qy=pqpx + qy = pq.
7 Marks

Question 5

14 MarksMedium
(a)
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}.
3 Marks
(b)
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.
4 Marks
(c)
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).
7 Marks
OR OPTION
(a)
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}.
3 Marks
(b)
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}).
4 Marks
(c)
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.
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) (Summer 2026, 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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