Question 1
(a)
Solve 𝑑𝑦 𝑑𝑥 + (𝑐𝑜𝑡𝑥)𝑦 = 2𝑐𝑜𝑠𝑥
3 Marks
(b)
Give Beta and Gamma function relationship and find 𝛽(9 2 , 7
2
4 Marks
(c)
Give the Statement of Convolution Theorem and find the Inverse Laplace transform of 𝑠+2 𝑠2(𝑠+3)
7 Marks
Gujarat Technological UniversitySummer 2025 Examination
GTU 2130002 Advance Engineering Mathematics (AEM) Summer 2025 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 Hours2:30 PM – 5:30 PM
Paper Structure5 QuestionsWith internal OR choices
Question 2
(a)
Check the Exactness of (𝑥3 + 3𝑥𝑦2)𝑑𝑥 + (3𝑥2𝑦 + 𝑦3)𝑑𝑦 = 0
3 Marks
(b)
Solve 𝑑𝑦 𝑑𝑥 + 2𝑦 𝑥 = 𝑥2𝑦2
4 Marks
(c)
Find the Fourier Series of 𝑓(𝑥) = {−𝜋 − 𝜋 < 𝑥 < 0 𝑥 0 < 𝑥 < 𝜋 }
7 Marks
OR OPTION
(c)
Solve the Initial –Value problem using Laplace transform 𝑦′′ + 3𝑦′ + 2𝑦 = 𝑒𝑡 , y(0)=1, 𝑦′(0) = 0
7 Marks
Question 3
(a)
Using the definition of Laplace transform prove that
1)𝐿(𝑡𝑛) = 𝑛!
𝑠𝑛+1
2)𝐿(𝑒−𝑎𝑡) = 1
𝑠−𝑎
3 Marks
(b)
Form the partial differential equation of 𝑧 = 𝑓(𝑥 𝑦)
4 Marks
(c)
Using Method of Variation of parameter solve (𝐷2 + 4)𝑦 = 𝑡𝑎𝑛2𝑥
7 Marks
OR OPTION
(a)
Find 𝐿(𝑡𝑒4𝑡𝑐𝑜𝑠2𝑡)
3 Marks
(b)
Using Partial differential equation eliminate the function 𝑓 from the relation 𝑓(𝑥𝑦 + 𝑧2,x+y+z)=0
4 Marks
(c)
Using Method of Undetermined coefficient solve (𝐷2 − 2𝐷)𝑦 = 𝑒𝑥(𝑠𝑖𝑛𝑥)
7 Marks
Question 4
(a)
Find the Fourier Sine series of 𝑓(𝑥) = 2𝑥 𝑖𝑛 0 < 𝑥 <
3 Marks
(b)
If 𝐿(𝑓(𝑡)) = log (𝑠+3 𝑠+1) find 𝐿(𝑓(2𝑡)) using change of scale property of Laplace transform.
4 Marks
(c)
Solve (𝑥2𝑦 − 2𝑥𝑦2)𝑑𝑥 − (𝑥3 − 3𝑥2𝑦)𝑑𝑦 = 0
7 Marks
OR OPTION
(a)
Find the Fourier Cosine integral of 𝑓(𝑥) = 𝜋 2 𝑒−𝑥 , 𝑥 ≥ 0
3 Marks
(b)
State First Shifting theorem of Laplace transform and using it find 𝐿(𝑒−3𝑡𝑡4)
4 Marks
(c)
Using Cauchy-Euler equation 𝑥2 𝑑2𝑦 𝑑𝑥2 − 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = sin (𝑙𝑜𝑔𝑥)
7 Marks
Question 5
(a)
Find the general solution to the partial differential equation 𝑥𝑝 + 𝑦𝑞 = 𝑥 − 𝑦
3 Marks
(b)
Solve 𝜕2𝑧 𝜕𝑥2 + 3 𝜕2𝑧 𝜕𝑥.𝜕𝑦 + 2 𝜕2𝑧 𝜕𝑦2 = 𝑥 + 𝑦
4 Marks
(c)
Discuss about ordinary point ,singular point and its types for the differential equation 𝑥3(𝑥 −
1)𝑦′′ + 3(𝑥 −
1)𝑦′ + 7𝑥𝑦 = 0
7 Marks
OR OPTION
(a)
Solve 𝑝(1 + 𝑞) = 𝑞𝑧
3 Marks
(b)
Solve 𝜕3𝑧 𝜕𝑥3 − 2 𝜕3𝑧 𝜕2𝑥.𝜕𝑦 = 2𝑒2𝑥
4 Marks
(c)
Find the Power Series solution of (1 + 𝑥2)𝑦′′ + 𝑥𝑦′ − 9𝑦 = 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 Advance Engineering Mathematics (AEM) (Summer 2025, 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.