Question 1
(a)
Solve 9𝑦𝑦′ + 4𝑥 = 0.
3 Marks
(b)
Solve the initial value problem 𝑦′ − (1 + 3𝑥−1)𝑦 = 𝑥 + 2; 𝑦(1) = 𝑒 − 1.
4 Marks
(c)
Find the Fourier series of the function 𝑓(𝑥) = 𝑥2; −𝜋 < 𝑥 < 𝜋.
7 Marks
Gujarat Technological UniversitySummer 2024 Examination
GTU 2130002 Advance Engineering Mathematics (AEM) Summer 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)
Find the general solution of 𝑦′′ + 3𝑦′ + 2𝑦 = 0.
3 Marks
(b)
Solve 𝑦′′′ − 3𝑦′′ + 3𝑦′ − 𝑦 = 4𝑒𝑡.
4 Marks
(c)
Using the method of variation of parameters find the general solution of (𝐷2 − 2𝐷 + 1)𝑦 = 3𝑥3 2𝑒𝑥.
7 Marks
OR OPTION
(c)
Using the method of undermined coefficients, find a particular solution of 𝑦′′ − 4𝑦′ − 12𝑦 = 8𝑥2.
7 Marks
Question 3
(a)
Obtain the Fourier series for the function 𝑓(𝑥) given by 𝑓(𝑥) = {1 + (2𝑥 𝜋 ) ; −𝜋 ≤ 𝑥 ≤ 0 1 − (2𝑥 𝜋 ) ; 0 ≤ 𝑥 ≤ 𝜋 . Hence, deduce that 1 12 + 1 32 + 1 52 + ⋯ = 𝜋2 8 .
7 Marks
(b)
A function 𝑓(𝑥) is defined by 𝑓(𝑥) = {1; −1 ≤ 𝑥 ≤ 1 0; 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Find the Fourier integral representation of 𝑓(𝑥). Hence, evaluate (a) ∫ 𝑠𝑖𝑛 λ𝑐𝑜𝑠𝑥 λ λ 𝑑 λ. ∞ 0 (b) ∫ 𝑠𝑖𝑛 λ λ 𝑑 λ. ∞ 0
7 Marks
OR OPTION
(a)
Find a cosine series of period 2𝜋 to represent 𝑓(𝑥) = 𝑠𝑖𝑛𝑥 in 0 < 𝑥 < 𝜋.
Also, graph the corresponding periodic continuation of 𝑓(𝑥). Hence deduce that 1 − 1 3 + 1 5 − 1 7 + ⋯ . = 𝜋
7 Marks
(b)
Determine the series solution for the differential equation 𝑦′′ + 𝑦 = 0 about 𝑥0 = 0.
7 Marks
Question 4
(a)
Find the Laplace transform of 𝑡3 + 𝑒−3𝑡 + 𝑡3
2
3 Marks
(b)
Find the Laplace transform of 𝑐𝑜𝑠ℎ(𝑘𝑡)𝑐𝑜𝑠𝑘𝑡
4 Marks
(c)
Find the inverse Laplace transform of 2𝑠+3 (𝑠+2)(𝑠+1)2
7 Marks
OR OPTION
(a)
Define
i)Gamma function and Beta function
ii)Write the relation between Beta and Gamma function.
3 Marks
(b)
Find the Laplace transform of unit step function 𝑓(𝑡) = {0; 0 ≤ 𝑡 < 𝑘 1; 𝑡 ≥ 𝑘
4 Marks
(c)
Solve the IVP using the Laplace transform:
𝑦′′ + 4𝑦 = 0; 𝑦(0) = 1, 𝑦′(0) = 6.
7 Marks
Question 5
(a)
Form the partial differential equation by eliminating the arbitrary constants for 𝑎𝑧 + 𝑏 = 𝑎2𝑥 + 𝑦.
3 Marks
(b)
Solve (𝑦 + 𝑧)𝑝 − (𝑥 + 𝑧)𝑞 = 𝑥 − 𝑦.
4 Marks
(c)
Using the method of separation of variables, solve 𝜕𝑢 𝜕𝑥 = 2 𝜕𝑢 𝜕𝑡 + 𝑢.
7 Marks
OR OPTION
(a)
Find the complete integral of 𝑧 = 𝑝𝑥 + 𝑞𝑦 + 𝑝𝑞.
3 Marks
(b)
Solve (𝐷2 + 10𝐷𝐷′ + 25𝐷′2 )𝑧 = 𝑒3𝑥+2𝑦.
4 Marks
(c)
The base of semi-infinite strip of metal plate is 30cm and is kept at 100°C. The two long edges are at zero temperature. Find the temperature at any point 15cm away from the base and situated midway between the long edges.
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) (Summer 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.