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Gujarat Technological UniversitySummer 2024 Examination

GTU 3130005 Complex Variables and Partial Differential Equations (CVPD) Summer 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)
Represent (1+i)21−i\frac{(1+i)^2}{1-i} in a+iba+ib or u+ivu+iv form and find its modulus and argument.
3 Marks
(b)
Define a harmonic function. Show that u(x,y)=x2−y2u(x, y) = x^2 - y^2 is harmonic and find the corresponding analytic function f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y).
4 Marks
(c)
Find the bilinear transformation which maps the points z=i,1,−iz = i, 1, -i onto the points w=−i,1,iw = -i, 1, i respectively.
7 Marks

Question 2

14 MarksMedium
(a)
Evaluate ∫04+2izˉ dz\int_0^{4+2i} \bar{z} \, dz along the curve z=t2+itz = t^2 + it.
3 Marks
(b)
Find the centre and radius of convergence of the given power series: ∑n=0∞(2n)!(n!)2(z−3i)n\sum_{n=0}^\infty \frac{(2n)!}{(n!)^2}(z - 3i)^n
4 Marks
(c)
Define analytic function and If f(z)f(z) is analytic and ∣f(z)∣=c|f(z)| = c then show that f(z)f(z) is constant.
7 Marks
OR OPTION
(c)
Evaluate ∮Ccos⁡πz2(z−1)(z−2) dz\oint_C \frac{\cos \pi z^2}{(z-1)(z-2)} \, dz; CC is ∣z∣=3|z| = 3.
7 Marks

Question 3

14 MarksMedium
(a)
Define: Singular point, Isolated singular point, Residue and Explain types of isolated singular points.
3 Marks
(b)
Find the Laurent series of 7z−2(z+1)z(z−2);1<∣z+1∣<3\frac{7z - 2}{(z+1)z(z-2)}; \quad 1 < |z+1| < 3
4 Marks
(c)
Evaluate using Cauchy residue theorem ∮C2z+6z2+4 dz\oint_C \frac{2z + 6}{z^2 + 4} \, dz where C:∣z−i∣=2C: |z - i| = 2.
7 Marks
OR OPTION
(a)
Find the pole and its order of following functions:
1f(z)=sin⁡zz4f(z) = \frac{\sin z}{z^4}
2f(z)=1(z−5)3(z2−4)f(z) = \frac{1}{(z-5)^3 (z^2 - 4)}
3 Marks
(b)
Find the residues at singular points of f(z)=z2(z−1)2(z+2)f(z) = \frac{z^2}{(z-1)^2 (z+2)}
4 Marks
(c)
Evaluate following real integration using residue theorem: ∫02π1(2+cos⁡θ)2 dθ\int_0^{2\pi} \frac{1}{(2 + \cos \theta)^2} \, d\theta
7 Marks

Question 4

14 MarksMedium
(a)
Form PDE by eliminating arbitrary Functions F(x+y+z,x2+y2−z2)=0F(x + y + z, x^2 + y^2 - z^2) = 0.
3 Marks
(b)
Solve following Linear Partial Differential Equations:
1xp+yq=x−yxp + yq = x - y
2(z−y)p+(x−z)q=y−x(z - y)p + (x - z)q = y - x
4 Marks
(c)
Solve following Non-linear partial differential equations using Charpit’s method: px+qy=pqpx + qy = pq.
7 Marks
OR OPTION
(a)
Find the order of the following PDE (1 to 3). And check whether the equations are linear, quasilinear or nonlinear:
1∂u∂x−∂u∂t=0\frac{\partial u}{\partial x} - \frac{\partial u}{\partial t} = 0
2∂2u∂x2−e2x∂2u∂t2=u3\frac{\partial^2 u}{\partial x^2} - e^{2x}\frac{\partial^2 u}{\partial t^2} = u^3
3uyuyy+(ux)2=0u_y u_{yy} + (u_x)^2 = 0
3 Marks
(b)
Solve following Non-linear partial differential equations:
1p2−q2=x−yp^2 - q^2 = x - y
2p(1+q)=qzp(1 + q) = qz
4 Marks
(c)
Solve following Partial Differential Equations:
1(x2−y2−z2)p+2xyq=2xz(x^2 - y^2 - z^2)p + 2xyq = 2xz
2∂2z∂x2−2∂2z∂x∂y−8∂2z∂y2=0\frac{\partial^2 z}{\partial x^2} - 2\frac{\partial^2 z}{\partial x \partial y} - 8\frac{\partial^2 z}{\partial y^2} = 0
7 Marks

Question 5

14 MarksMedium
(a)
Classify second order homogeneous partial differential equations as elliptic, parabolic or hyperbolic:
1∂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
2∂2z∂x2−∂2z∂x∂y=0\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial x \partial y} = 0
3 Marks
(b)
Using method of separation of variables solve ∂u∂t+u=2∂u∂x\frac{\partial u}{\partial t} + u = 2\frac{\partial u}{\partial x} given that u(x,0)=4e−3xu(x, 0) = 4e^{-3x}.
4 Marks
(c)
Determine solution of two-dimensional Laplace equation ∂2u∂x2+∂2u∂y2=0\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 which satisfy the condition u(0,y)=u(l,y)=u(x,0)u(0, y) = u(l, y) = u(x, 0) and u(x,b)=sin⁡nπxlu(x, b) = \sin \frac{n\pi x}{l}.
7 Marks
OR OPTION
(a)
Solve partial differential equations by direct integration method: ∂2z∂x2=sin⁡x\frac{\partial^2 z}{\partial x^2} = \sin x.
3 Marks
(b)
Solve second order homogeneous PDEs: ∂2z∂x2−∂2z∂x∂y−12∂2z∂y2=3e2x−3y\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial x \partial y} - 12\frac{\partial^2 z}{\partial y^2} = 3e^{2x - 3y}
4 Marks
(c)
Find the solution of One-Dimensional Wave Equation.
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 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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