Q.1
14 Marks(a)
Solve 9𝑦𝑦′ + 4𝑥 = 0.
(b)
Solve the initial value problem 𝑦′ − (1 + 3𝑥⁻¹)𝑦 = 𝑥 + 2; 𝑦(1) = 𝑒 − 1.
(c)
Find the Fourier series of the function 𝑓(𝑥) = 𝑥²; −𝜋 < 𝑥 < 𝜋.
Gujarat Technological UniversitySummer 2024 examination
GTU 2130002 Advance Engineering Mathematics (AEM) Summer 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
No solutions on this paper yetNeed them for your exam? Ask once. Papers that students ask for are solved first, and we aim to add them within 24 hours.
Noted. This paper is in our queue now. Check back tomorrow.
We could not record that here. Email the paper name to hirenjsca@gmail.com and we will add it.
Q.2
14 Marks(a)
Find the general solution of 𝑦′′ + 3𝑦′ + 2𝑦 = 0.
(b)
Solve 𝑦′′′ − 3𝑦′′ + 3𝑦′ − 𝑦 = 4𝑒𝑡.
(c)
Using the method of variation of parameters find the general solution of (𝐷² − 2𝐷 + 1)𝑦 = 3𝑥³ 2𝑒𝑥.
OR
(c)
Using the method of undermined coefficients, find a particular solution of 𝑦′′ − 4𝑦′ − 12𝑦 = 8𝑥².
Q.3
14 Marks(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 .
(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
OR
(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
(b)
Determine the series solution for the differential equation 𝑦′′ + 𝑦 = 0 about 𝑥0 = 0.
Q.4
14 Marks(a)
Find the Laplace transform of 𝑡³ + 𝑒−3𝑡 + 𝑡3
2
(b)
Find the Laplace transform of 𝑐𝑜𝑠ℎ(𝑘𝑡)𝑐𝑜𝑠𝑘𝑡
(c)
Find the inverse Laplace transform of 2𝑠+3 (𝑠+2)(𝑠+1)²
OR
(a)
Define
i)Gamma function and Beta function
ii)Write the relation between Beta and Gamma function.
(b)
Find the Laplace transform of unit step function 𝑓(𝑡) = {0; 0 ≤ 𝑡 < 𝑘 1; 𝑡 ≥ 𝑘
(c)
Solve the IVP using the Laplace transform:
𝑦′′ + 4𝑦 = 0; 𝑦(0) = 1, 𝑦′(0) = 6.
Q.5
14 Marks(a)
Form the partial differential equation by eliminating the arbitrary constants for 𝑎𝑧 + 𝑏 = 𝑎²𝑥 + 𝑦.
(b)
Solve (𝑦 + 𝑧)𝑝 − (𝑥 + 𝑧)𝑞 = 𝑥 − 𝑦.
(c)
Using the method of separation of variables, solve 𝜕𝑢 𝜕𝑥 = 2 𝜕𝑢 𝜕𝑡 + 𝑢.
OR
(a)
Find the complete integral of 𝑧 = 𝑝𝑥 + 𝑞𝑦 + 𝑝𝑞.
(b)
Solve (𝐷² + 10𝐷𝐷′ + 25𝐷′2 )𝑧 = 𝑒3𝑥+2𝑦.
(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.
More Advance Engineering Mathematics papers
Questions repeat across sessions. Working through two or three papers shows you which topics GTU asks most.
Other Semester 3 subjects
BE Common
Preparing with friends?
Send this paper to your class group. Every question has its own link.
About this paper
Official Gujarat Technological University (GTU) examination paper 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.