Q.1
14 Marks(a)
Solve: 𝑥2𝑑𝑦 + 𝑦 (𝑥 + 𝑦)𝑑𝑥 = 0
(b)
Find the Laplace transform of 𝑡² sin 𝜋𝑡
(c)
Find a Fourier series for a periodic function f(x) with period 2, where 𝑓(𝑥) = −1, −1 < 𝑥 < 0 = 1, 0 < 𝑥 <
Gujarat Technological UniversityWinter 2024 examination
GTU 2130002 Advance Engineering Mathematics (AEM) Winter 2024 Question Paper & PDF
B.E. · Common Engineering · Semester 3 · Subject code 2130002
- Total
- 70 marks
- To pass
- 23 marks (33%)
- Duration
- 2.5 Hours 10:30 AM – 1:30 PM
- Questions
- 5 with internal choice
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Q.2
14 Marks(a)
Define Beta function, Gamma function and write the relation between Beta and Gamma function.
(b)
Solve: (𝑥 +
1)𝑑𝑦
𝑑𝑥 − 𝑦 = 𝑒3𝑥(𝑥 +
1)2
(c)
Find the Fourier series of 𝑓(𝑥) = 𝑥 + |𝑥|, −𝜋 < 𝑥 < 𝜋 .
OR
(c)
Solve 𝑦′′ + 4𝑦 = 8𝑥² by using the method of undetermined coefficients.
Q.3
14 Marks(a)
Laplace Find 𝐿−1 {𝑠³+2𝑠²+2 𝑠3(𝑠²+1) }
(b)
Find particular integral for an equation (𝐷² − 4𝐷 + 3)𝑦 = sin 3𝑥 cos 2𝑥
(c)
Solve the IVP 𝑦′′ − 2𝑦′ = 𝑒𝑡 sin 𝑡 , 𝑦(0) = 𝑦′(0) = 0
OR
(a)
By using first shifting theorem find 𝐿{(𝑡 + 1)²𝑒𝑡}
(b)
Solve 𝑑2𝑦 𝑑𝑥2 + 4𝑦 = tan 2𝑥 by using Variation of Parameter.
(c)
Using convolution theorem find the inverse transform of 𝑎 𝑠2(𝑠²+𝑎2)
Q.4
14 Marks(a)
Solve 𝑑𝑦 𝑑𝑥 + 2𝑦 𝑥 = sin 𝑥
(b)
Find the Laplace Transform of 𝑒−2𝑡(sin 4𝑡 + 𝑡²)
(c)
Find the Power series solution of the equation (𝑥² + 1)𝑦′′ + 𝑥𝑦′ − 𝑥𝑦 = 0
OR
(a)
IFind the Laplace Transform of cos 𝑎𝑡−𝑐𝑜𝑠𝑏𝑡 𝑡
IIFind the Laplace transform of 𝐿{𝑡² 𝑢(𝑡 − 2)}
i)Find the Laplace Transform of cos 𝑎𝑡−𝑐𝑜𝑠𝑏𝑡
𝑡
ii)Find the Laplace transform of 𝐿{𝑡² 𝑢(𝑡 −
2)}
(b)
Find half-range cosine series for 𝑓(𝑥) = (𝑥 − 1)², 0 < 𝑥 <
(c)
Find the series solution of 𝑥𝑦′′ + 𝑦′ + 𝑥𝑦 = 0
Q.5
14 Marks(a)
IFind the complete integral of 𝑝𝑞 = 4𝑧
IIFind the complete integral of 𝑝 − 𝑥² = 𝑞 + 𝑦²
i)Find the complete integral of 𝑝𝑞 = 4𝑧
ii)Find the complete integral of 𝑝 − 𝑥² = 𝑞 + 𝑦²
(b)
Form a partial differential equation by eliminating arbitrary constants a and b from equation 𝑧 = (𝑥² + 𝑎)(𝑦² + 𝑏)
(c)
Using Charpti’s method solve: 𝑧 = 𝑝𝑞
OR
(a)
Form a partial differential equation by eliminating arbitrary function from ∅( 𝑥² − 𝑦², 𝑥𝑦𝑧) = 0
(b)
Solve 𝑥2𝑝 + 𝑦²𝑞 = 𝑧²
(c)
Solve the equation 𝜕𝑢 𝜕𝑥 = 2 𝜕𝑢 𝜕𝑡 + 𝑢, given 𝑢(𝑥, 0) = 4𝑒−4𝑥 by using Separation of variable method.
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About this paper
Official Gujarat Technological University (GTU) examination paper for Advance Engineering Mathematics (AEM) (Winter 2024, 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.