Question 1
(a)
Define Beta and Gamma functions. Also prove that ∫ 𝑥𝑑𝑥 √1−𝑥5 1 0 = 1 5 𝛽 (2 5 , 1
2.
7 Marks
(b)
Find the Fourier series expansion for 𝑓(𝑥) = 𝑥2, 0 ≤ 𝑥 ≤ 2𝜋.
7 Marks
Gujarat Technological UniversitySummer 2025 Examination
GTU 130001 Mathematics-III (Maths 3) Summer 2025 Paper Solution & PDF
B.E. · Computer Engineering · Semester 3 · Subject Code: 130001
Total Marks70 MarksExternal theory exam
Passing Marks23 Marks33% minimum cutoff
Exam Duration2.5 Hours2:30 PM – 5:30 PM
Paper Structure5 QuestionsWith internal OR choices
Question 2
(a)
Obtain the Fourier series of periodic function 𝑓(𝑥) = 2𝑥, −1 < 𝑥 < 1, 𝑝 = 2.
7 Marks
(b)
Test for exactness and solve [(𝑥 + 1)𝑒𝑥 − 𝑒𝑦]𝑑𝑥 − 𝑥𝑒𝑦𝑑𝑦 = 0, 𝑦(1) = 0.
7 Marks
OR OPTION
(b)
Find the solution of differential equation 𝑑𝑦 𝑑𝑥 + 𝑦 = − 𝑥 𝑦.
7 Marks
Question 3
(a)
Solve the initial value problem 𝑦′′ + 𝑦′ − 2𝑦 = 0, 𝑦(0) = 4 𝑎𝑛𝑑 𝑦′(0) = −5.
7 Marks
(b)
Find the solution of differential equation 𝑦′′ + 4𝑦 = 2𝑠𝑖𝑛3𝑥 by the method of undetermined co-efficients.
7 Marks
OR OPTION
(a)
Using the method of variation of parameter, solve 𝑦′′ + 𝑦 = 𝑠𝑒𝑐𝑥.
7 Marks
(b)
Solve (𝐷2 + 4𝐷 + 4)𝑦 = 𝑒−2𝑥 𝑥2 .
7 Marks
Question 4
(a)
Find Laplace transforms of
i)𝑠𝑖𝑛ℎ𝑎𝑡𝑠𝑖𝑛𝑎𝑡
ii)𝑡𝑒𝑡𝑠𝑖𝑛𝑡
7 Marks
(b)
Using Convolution theorem find 𝐿−1 [ 1 𝑠(𝑠2+4)]
7 Marks
OR OPTION
(a)
(
1)Find 𝐿−1 [ 5𝑠2+3𝑠−16
(𝑠−1)(𝑠−2)(𝑠+3)]
2)Find 𝐿 {∫ 𝑒−𝑢𝑐𝑜𝑠𝑢𝑑𝑢
𝑡
0 }
7 Marks
(b)
Solve the equation 𝑑2𝑦 𝑑𝑥2 + 𝑦 = 0 by the power series method.
7 Marks
Question 5
(a)
A rod of length L with insulated side is initially at uniform temperature 100. Its ends are suddenly cooled at 0𝑜𝑐 and kept at that temperature. Find the temperature function 𝑢(𝑥, 𝑡).
7 Marks
(b)
Prove that ∫ 1−𝑐𝑜𝑠𝜋𝑤 𝑤 ∞ 0 𝑠𝑖𝑛𝑤𝑥𝑑𝑤 = { 𝜋 2 , 𝑖𝑓 0 < 𝑥 < 𝜋 0, 𝑖𝑓 𝑥 > 𝜋.
7 Marks
OR OPTION
(a)
Solve (𝑥2𝐷2 − 3𝑥𝐷 + 4)𝑦 = 0, 𝑦(1) = 0, 𝑦′(1) = 3.
7 Marks
(b)
Find Fourier cosine integral of 𝑓(𝑥) = 𝑒−𝑘𝑥, where 𝑥 > 0, 𝑘 > 0.
7 Marks
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About this Examination Paper & Attribution
Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Mathematics-III (Maths 3) (Summer 2025, B.E. · Computer 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.