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Gujarat Technological UniversityWinter 2024 Examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Winter 2024 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)
Find the principal argument Arg(z)Arg(z) when z=−21+i3z = \frac{-2}{1+i\sqrt{3}}.
3 Marks
(b)
Find all the values of following: (1−i)23(1-i)^{\frac{2}{3}}.
4 Marks
(c)
Find the bilinear transformation which maps the points z=0,−i,−1z = 0, -i, -1 into w=i,1,0w = i, 1, 0 respectively.
7 Marks

Question 2

14 MarksMedium
(a)
Show that f(z)=z3f(z) = z^3 is analytic everywhere.
3 Marks
(b)
Evaluate ∫(x2−iy2) dz\int (x^2 - iy^2) \, dz along the parabola y=2x2y = 2x^2 from (1,2)(1,2) to (2,8)(2,8).
4 Marks
(c)
Find Laurent's series expansion in powers of zz that represent f(z)=1z2(1−z)f(z) = \frac{1}{z^2(1-z)} for the following domains: (i) ∣z∣<1|z| < 1 (ii) ∣z∣>1|z| > 1
7 Marks
OR OPTION
(c)
Find the image of the half-plane x>cx > c, when c>0c > 0 under the transformation w=1zw = \frac{1}{z}. Show the regions graphically.
7 Marks

Question 3

14 MarksMedium
(a)
If u+ivu + iv is analytic, show that v−iuv - iu and −v+iu-v + iu are also analytic.
3 Marks
(b)
Show that u(x,y)=x2−y2+xu(x,y) = x^2 - y^2 + x is harmonic. Find the corresponding analytic function f(z)=u+ivf(z) = u + iv.
4 Marks
(c)
Evaluate ∫Ccos⁡πzz2−1 dz\int_C \frac{\cos \pi z}{z^2 - 1} \, dz, where CC is the rectangle whose vertices are 2±i,−2±i2 \pm i, -2 \pm i.
7 Marks
OR OPTION
(a)
Obtain the residue of f(z)=z−3(z+1)(z+2)f(z) = \frac{z - 3}{(z + 1)(z + 2)} at its poles.
3 Marks
(b)
Evaluate ∫Ce2z(z+1)4 dz\int_C \frac{e^{2z}}{(z + 1)^4} \, dz, where CC is the circle ∣z∣=2|z| = 2.
4 Marks
(c)
Evaluate ∫0∞xsin⁡xx2+9 dx\int_0^{\infty} \frac{x \sin x}{x^2 + 9} \, dx using residue.
7 Marks

Question 4

14 MarksMedium
(a)
Form a partial differential equation for the equation (x−a)(y−b)−z2=x2+y2(x-a)(y-b) - z^2 = x^2 + y^2.
3 Marks
(b)
Solve: xp+yq=3zxp + yq = 3z.
4 Marks
(c)
Solve by Charpit's Method: p=(z+qy)2p = (z + qy)^2.
7 Marks
OR OPTION
(a)
Form a partial differential equation by eliminating the arbitrary function from z=f(x2−y2)z = f(x^2 - y^2).
3 Marks
(b)
Solve x(y−z)p+y(z−x)q=z(x−y)x(y - z)p + y(z - x)q = z(x - y).
4 Marks
(c)
Solve (D2−2DD′+D′2)z=ex+2y+x3(D^2 - 2DD' + D'^2)z = e^{x+2y} + x^3.
7 Marks

Question 5

14 MarksMedium
(a)
Solve: p−x2=q+y2p - x^2 = q + y^2.
3 Marks
(b)
Solve (D2−2DD′+D′2)z=tan⁡(x+y)(D^2 - 2DD' + D'^2)z = \tan(x + y).
4 Marks
(c)
Solve (D2+DD′−6D′2)z=sin⁡(2x+y)(D^2 + DD' - 6D'^2)z = \sin(2x + y).
7 Marks
OR OPTION
(a)
Solve: (p−q)(z−px−qy)=1(p - q)(z - px - qy) = 1.
3 Marks
(b)
Solve: ∂3z∂x3−3∂3z∂x2∂y+2∂3z∂y3=0\frac{\partial^3 z}{\partial x^3} - 3 \frac{\partial^3 z}{\partial x^2 \partial y} + 2 \frac{\partial^3 z}{\partial y^3} = 0.
4 Marks
(c)
Using the method of separation of variables, solve ∂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 2024, 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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