Question 1
(a)
Solve: 𝑥2𝑑𝑦 + 𝑦 (𝑥 + 𝑦)𝑑𝑥 = 0
3 Marks
(b)
Find the Laplace transform of 𝑡2 sin 𝜋𝑡
4 Marks
(c)
Find a Fourier series for a periodic function f(x) with period 2, where 𝑓(𝑥) = −1, −1 < 𝑥 < 0 = 1, 0 < 𝑥 <
7 Marks
Gujarat Technological UniversityWinter 2024 Examination
GTU 2130002 Advance Engineering Mathematics (AEM) Winter 2024 Paper Solution & PDF
B.E. · Common Engineering · Semester 3 · Subject Code: 2130002
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
Question 2
(a)
Define Beta function, Gamma function and write the relation between Beta and Gamma function.
3 Marks
(b)
Solve: (𝑥 +
1)𝑑𝑦
𝑑𝑥 − 𝑦 = 𝑒3𝑥(𝑥 +
1)2
4 Marks
(c)
Find the Fourier series of 𝑓(𝑥) = 𝑥 + |𝑥|, −𝜋 < 𝑥 < 𝜋 .
7 Marks
OR OPTION
(c)
Solve 𝑦′′ + 4𝑦 = 8𝑥2 by using the method of undetermined coefficients.
7 Marks
Question 3
(a)
Laplace Find 𝐿−1 {𝑠3+2𝑠2+2 𝑠3(𝑠2+1) }
3 Marks
(b)
Find particular integral for an equation (𝐷2 − 4𝐷 + 3)𝑦 = sin 3𝑥 cos 2𝑥
4 Marks
(c)
Solve the IVP 𝑦′′ − 2𝑦′ = 𝑒𝑡 sin 𝑡 , 𝑦(0) = 𝑦′(0) = 0
7 Marks
OR OPTION
(a)
By using first shifting theorem find 𝐿{(𝑡 + 1)2𝑒𝑡}
3 Marks
(b)
Solve 𝑑2𝑦 𝑑𝑥2 + 4𝑦 = tan 2𝑥 by using Variation of Parameter.
4 Marks
(c)
Using convolution theorem find the inverse transform of 𝑎 𝑠2(𝑠2+𝑎2)
7 Marks
Question 4
(a)
Solve 𝑑𝑦 𝑑𝑥 + 2𝑦 𝑥 = sin 𝑥
3 Marks
(b)
Find the Laplace Transform of 𝑒−2𝑡(sin 4𝑡 + 𝑡2)
4 Marks
(c)
Find the Power series solution of the equation (𝑥2 + 1)𝑦′′ + 𝑥𝑦′ − 𝑥𝑦 = 0
7 Marks
OR OPTION
(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)}
3 Marks
(b)
Find half-range cosine series for 𝑓(𝑥) = (𝑥 − 1)2, 0 < 𝑥 <
4 Marks
(c)
Find the series solution of 𝑥𝑦′′ + 𝑦′ + 𝑥𝑦 = 0
7 Marks
Question 5
(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
3 Marks
(b)
Form a partial differential equation by eliminating arbitrary constants a and b from equation 𝑧 = (𝑥2 + 𝑎)(𝑦2 + 𝑏)
4 Marks
(c)
Using Charpti’s method solve: 𝑧 = 𝑝𝑞
7 Marks
OR OPTION
(a)
Form a partial differential equation by eliminating arbitrary function from ∅( 𝑥2 − 𝑦2, 𝑥𝑦𝑧) = 0
3 Marks
(b)
Solve 𝑥2𝑝 + 𝑦2𝑞 = 𝑧2
4 Marks
(c)
Solve the equation 𝜕𝑢 𝜕𝑥 = 2 𝜕𝑢 𝜕𝑡 + 𝑢, given 𝑢(𝑥, 0) = 4𝑒−4𝑥 by using Separation of variable method.
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 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.