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Gujarat Technological UniversitySummer 2023 Examination

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

Question 1

14 MarksMedium
(a)
Determine an analytic function whose real part is e2x(xcos⁡2y−ysin⁡2y)e^{2x}(x \cos 2y - y \sin 2y).
3 Marks
(b)
Solve the equation z2+(2i−3)z+5−i=0z^2 + (2i - 3)z + 5 - i = 0.
4 Marks
(c)
Show that u(x,y)=2x−x3+3xy2u(x, y) = 2x - x^3 + 3xy^2 is harmonic and find a harmonic conjugate of u(x,y)u(x, y).
7 Marks

Question 2

14 MarksMedium
(a)
Evaluate ∫C∣z∣2 dz\int_C |z|^2 \, dz around the square with vertices at (0,0),(1,0),(1,1),(0,1)(0, 0), (1, 0), (1, 1), (0, 1).
3 Marks
(b)

Expand f(z)=1(z+2)(z+4)f(z) = \frac{1}{(z+2)(z+4)} valid for the following regions:

i∣z∣<2|z| < 2
ii2<∣z∣<42 < |z| < 4
4 Marks
(c)
iEvaluate ∫Cz dz(z−1)(z−2)\int_C \frac{z \, dz}{(z-1)(z-2)} where CC is the circle ∣z∣=12|z| = \frac{1}{2}.
iiEvaluate ∫Cdzz2−7z+12\int_C \frac{dz}{z^2 - 7z + 12} where CC is the circle ∣z∣=3.5|z| = 3.5.
7 Marks
OR OPTION
(c)
Define mobius transformation. Determine the mobius transformation which maps z1=0,z2=1,z3=∞z_1 = 0, z_2 = 1, z_3 = \infty onto w1=−1,w2=−i,w3=1w_1 = -1, w_2 = -i, w_3 = 1.
7 Marks

Question 3

14 MarksMedium
(a)
Find and plot the image of triangular region in the zz-plane with vertices (0,0),(1,0),(0,1)(0, 0), (1, 0), (0, 1) under the transformation w=(1−i)z+3w = (1 - i)z + 3.
3 Marks
(b)
Find the values of aa and bb such that the function f(z)=x2+ay2−2xy+i(bx2−y2+2xy)f(z) = x^2 + ay^2 - 2xy + i(bx^2 - y^2 + 2xy) is analytic.
4 Marks
(c)
Determine the poles of the function f(z)=z2(z−1)2(z+2)f(z) = \frac{z^2}{(z-1)^2 (z+2)} and residue at each pole. Hence evaluate ∫Cf(z) dz\int_C f(z) \, dz where CC is the circle ∣z∣=3|z| = 3.
7 Marks
OR OPTION
(a)
Expand f(z)=1−ezzf(z) = \frac{1 - e^z}{z} in Laurent’s series about z=0z = 0.
3 Marks
(b)
Find modulus and argument of:
i1+2i1−(1−i)2\frac{1 + 2i}{1 - (1-i)^2}
ii(1+i)21−i\frac{(1+i)^2}{1-i}
4 Marks
(c)
iEvaluate ∫C3z2+7z+1z+1 dz\int_C \frac{3z^2 + 7z + 1}{z+1} \, dz where CC is ∣z∣=12|z| = \frac{1}{2}.
iiEvaluate ∫Cz2+1z2−1 dz\int_C \frac{z^2 + 1}{z^2 - 1} \, dz where CC is ∣z−1∣=1|z - 1| = 1.
7 Marks

Question 4

14 MarksMedium
(a)
Solve yzp−xzq=xyyz p - xz q = xy.
3 Marks
(b)
Form partial differential equation by eliminating the arbitrary constants aa and bb from z=axey+12a2e2y+bz = a x e^y + \frac{1}{2} a^2 e^{2y} + b.
4 Marks
(c)
iSolve 25r−40s+16t=025r - 40s + 16t = 0.
iiSolve p2+q2=x+yp^2 + q^2 = x + y.
7 Marks
OR OPTION
(a)
Solve (mz−ny)p+(nx−lz)q=ly−mx(mz - ny)p + (nx - lz)q = ly - mx.
3 Marks
(b)
Form a partial differential equation by eliminating the arbitrary functions from f(x2−y2,xyz)=0f(x^2 - y^2, xyz) = 0.
4 Marks
(c)
iSolve (D2−DD′+D′−1)z=cos⁡(x+2y)(D^2 - DD' + D' - 1)z = \cos(x + 2y).
iiSolve using Charpit’s Method: z2=pqxyz^2 = pqxy.
7 Marks

Question 5

14 MarksMedium
(a)
Solve (D2+10DD′+25D′2)z=e3x+2y(D^2 + 10DD' + 25D'^2)z = e^{3x+2y}.
3 Marks
(b)
Solve x∂u∂x−2y∂u∂y=0x \frac{\partial u}{\partial x} - 2y \frac{\partial u}{\partial y} = 0 using method of separation of variables.
4 Marks
(c)
iSolve (D2−D′2)z=x−y(D^2 - D'^2)z = x - y.
iiSolve (2D2−5DD′+2D′2)z=sin⁡(2x+y)(2D^2 - 5DD' + 2D'^2)z = \sin(2x + y).
7 Marks
OR OPTION
(a)
Solve (1−x)p+(z−y)q=3−z(1 - x)p + (z - y)q = 3 - z.
3 Marks
(b)
Solve 2∂u∂x=∂u∂t+u2\frac{\partial u}{\partial x} = \frac{\partial u}{\partial t} + u using method of separation of variables subject to the condition u(x,0)=4e−3xu(x, 0) = 4e^{-3x}.
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
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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 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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