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Gujarat Technological UniversityWinter 2025 Examination

GTU 2130002 Advance Engineering Mathematics (AEM) Winter 2025 Paper Solution & PDF

B.E. · Common Engineering · Semester 3 · Subject Code: 2130002
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Total Marks70 MarksExternal theory exam
Passing Marks23 Marks33% minimum cutoff
Exam Duration2.5 Hours10:30 AM – 1:30 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Define Signum function, Dirac's Delta function and Beta function.
3 Marks
(b)
Solve d3ydx3+dydx=cosec⁡x\frac{d^3 y}{dx^3} + \frac{dy}{dx} = \operatorname{cosec} x by method of variation of parameters.
4 Marks
(c)
Obtain the Fourier series of f(x)=(π−x2)2f(x) = \left(\frac{\pi - x}{2}\right)^2 in the interval 0≤x≤2π0 \le x \le 2\pi. Hence deduce that π212=112−122+132−…\frac{\pi^2}{12} = \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \dots
7 Marks

Question 2

14 MarksMedium
(a)
Find half range cosine series for f(x)=x,0<x<3f(x) = x, 0 < x < 3.
3 Marks
(b)
Solve d3ydx3−d2ydx2+3dydx+5y=excos⁡3x\frac{d^3 y}{dx^3} - \frac{d^2 y}{dx^2} + 3\frac{dy}{dx} + 5y = e^x \cos 3x.
4 Marks
(c)
Find the series solution of y′′=2y′y'' = 2y' in powers of xx.
7 Marks
OR OPTION
(c)
Find the series solution of (1+x2)y′′+xy′−xy=0(1 + x^2)y'' + xy' - xy = 0.
7 Marks

Question 3

14 MarksMedium
(a)
Solve dydx+2ytan⁡x=sin⁡x\frac{dy}{dx} + 2y\tan x = \sin x.
3 Marks
(b)
Solve (x3+3xy2)dx+(3x2y+y3)dy=0(x^3 + 3xy^2)dx + (3x^2 y + y^3)dy = 0.
4 Marks
(c)
Solve (D3−D2−6D)y=x2+1(D^3 - D^2 - 6D)y = x^2 + 1.
7 Marks
OR OPTION
(a)
Solve dydx+ycos⁡x+sin⁡y+ysin⁡x+xcos⁡y+x=0\frac{dy}{dx} + \frac{y\cos x + \sin y + y}{\sin x + x\cos y + x} = 0.
3 Marks
(b)
Solve (x2y2+2)y dx+(2−x2y2)x dy=0(x^2 y^2 + 2)y \, dx + (2 - x^2 y^2)x \, dy = 0.
4 Marks
(c)
Solve (D4+2D3−3D2)y=x2+3e2x(D^4 + 2D^3 - 3D^2)y = x^2 + 3e^{2x}.
7 Marks

Question 4

14 MarksMedium
(a)
Find the Laplace Transform of e−2t(sin⁡4t+t2)e^{-2t}(\sin 4t + t^2).
3 Marks
(b)
Find the Inverse Laplace Transform of log⁡(1+ω2s2)\log\left(1 + \frac{\omega^2}{s^2}\right).
4 Marks
(c)
Find the Inverse Laplace transform of 5s+3(s−1)(s2+2s+5)\frac{5s + 3}{(s - 1)(s^2 + 2s + 5)} using partial fractions.
7 Marks
OR OPTION
(a)
Find the Laplace Transform of sin⁡2tsin⁡3t\sin 2t \sin 3t.
3 Marks
(b)
Find the Laplace Transform of t2cosh⁡πtt^2 \cosh \pi t.
4 Marks
(c)
Solve the initial value problem d2xdt2+2dxdt+5x=e−tsin⁡t\frac{d^2 x}{dt^2} + 2\frac{dx}{dt} + 5x = e^{-t}\sin t where x(0)=0x(0) = 0 and x′(0)=1x'(0) = 1 using Laplace transform.
7 Marks

Question 5

14 MarksMedium
(a)
Form a partial differential equation by eliminating the arbitrary constants for the equation z=(x−2)2+(y−3)2z = (x - 2)^2 + (y - 3)^2.
3 Marks
(b)
Solve pz−qz=z2+(x+y)2pz - qz = z^2 + (x + y)^2 using Lagrange's method.
4 Marks
(c)
Solve ∂2z∂x2−2∂z∂x+∂z∂y=0\frac{\partial^2 z}{\partial x^2} - 2\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0 using method of separation of variables.
7 Marks
OR OPTION
(a)
Solve p(1+q)=qzp(1 + q) = qz.
3 Marks
(b)
Solve the equation by method of direct integration: ∂2z∂x∂y=sin⁡xsin⁡y\frac{\partial^2 z}{\partial x \partial y} = \sin x \sin y, given that ∂z∂y=−2sin⁡y\frac{\partial z}{\partial y} = -2\sin y when x=0x = 0 and z=0z = 0 when yy is an odd multiple of π2\frac{\pi}{2}.
4 Marks
(c)
Solve ∂u∂x+∂u∂y=2(x+y)u\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 2(x + y)u using method of separation of variables.
7 Marks
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About this Examination Paper & Attribution

Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Advance Engineering Mathematics (AEM) (Winter 2025, 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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