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Gujarat Technological UniversityWinter 2025 Examination

GTU 3110015 Mathematics - 2 (Maths 2) Winter 2025 Paper Solution & PDF

B.E. · Common Engineering · Semester 1 · Subject Code: 3110015
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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
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
If ϕ\phi is a scalar field, prove that curl(grad ϕ)=0\text{curl}(\text{grad } \phi) = 0.
3 Marks
(b)
Determine whether the vector field u⃗=y2i^+2xyj^−z2k^\vec{u} = y^2 \hat{i} + 2xy \hat{j} - z^2 \hat{k} is solenoidal or irrotational at the point (1,2,1)(1, 2, 1)?
4 Marks
(c)
Verify Green’s theorem in the plane for F⃗=(x2+y2)i^−2xyj^\vec{F} = (x^2 + y^2)\hat{i} - 2xy\hat{j} taken around the rectangle bounded by the lines x=±1,y=0x = \pm 1, y = 0 and y=2y = 2.
7 Marks

Question 2

14 MarksMedium
(a)
Define order and degree of the ordinary differential equation. Give an example of ordinary differential equation:
ifirst order first degree, (ii) first order higher degree.
3 Marks
(b)
Find the Laplace transform of:
itcos⁡2tt \cos^2 t
iisin⁡tt\frac{\sin t}{t}
4 Marks
(c)
Apply the method of undetermined coefficients to solve the differential equation (D2−2D+5)y=5x3−6x2+6x(D^2 - 2D + 5)y = 5x^3 - 6x^2 + 6x
7 Marks
OR OPTION
(c)
Apply the variation of parameters method to solve the differential equation (D2−3D+2)y=ex(D^2 - 3D + 2)y = e^x
7 Marks

Question 3

14 MarksMedium
(a)
State first shifting theorem for Laplace transform. Find the Laplace transform of e−3t(t2+sin⁡t)e^{-3t}(t^2 + \sin t).
3 Marks
(b)
Find the general solution of the differential equation x2y dx−(x3+xy2) dy=0x^2 y \, dx - (x^3 + xy^2) \, dy = 0.
4 Marks
(c)
Apply Laplace transform to solve the initial value problem:

y′′−6y′+9y=t2e3t,y(0)=2,  y′(0)=6y'' - 6y' + 9y = t^2 e^{3t}, \quad y(0) = 2, \; y'(0) = 6

7 Marks
OR OPTION
(a)
State convolution theorem for Laplace transform. Find the inverse Laplace transform of 1s(s+1)\frac{1}{s(s+1)}
3 Marks
(b)
Check whether the given differential equation is exact or not:

(x4−2xy2+y4) dx−(2x2y−4xy3+sin⁡y) dy=0(x^4 - 2xy^2 + y^4) \, dx - (2x^2 y - 4xy^3 + \sin y) \, dy = 0

4 Marks
(c)
Find the Fourier cosine integral representation of f(x)=π2e−x,x>0f(x) = \frac{\pi}{2} e^{-x}, \quad x > 0
7 Marks

Question 4

14 MarksMedium
(a)
Find the general solution of the differential equation: y′+2ytan⁡x=sin⁡xy' + 2y \tan x = \sin x.
3 Marks
(b)
Solve the Euler-Cauchy differential equation: (4x2D2+16xD+9)y=0(4x^2 D^2 + 16xD + 9)y = 0.
4 Marks
(c)
Solve:
i(D2−4)y=e2x+e−4x(D^2 - 4)y = e^{2x} + e^{-4x}
ii(D3−3D2−D+3)y=1(D^3 - 3D^2 - D + 3)y = 1
7 Marks
OR OPTION
(a)
Solve: x2p2+xyp−6y2=0x^2 p^2 + xyp - 6y^2 = 0.
3 Marks
(b)
If y1=xy_1 = x is one solution of x2y′′+xy′−y=0x^2 y'' + xy' - y = 0, find the second solution.
4 Marks
(c)
Solve:
i(D2+9)y=cos⁡4x(D^2 + 9)y = \cos 4x
ii(D3−5D2+8D−4)y=e−x(D^3 - 5D^2 + 8D - 4)y = e^{-x}
7 Marks

Question 5

14 MarksMedium
(a)
Classify the singular points of the differential equation x2y′′+xy′−2y=0x^2 y'' + xy' - 2y = 0.
3 Marks
(b)
Find the series solution of y′′=2y′y'' = 2y' in powers of xx.
4 Marks
(c)
Using the power-series method, solve (1−x2)y′′−2xy′+2y=0(1 - x^2)y'' - 2xy' + 2y = 0.
7 Marks
OR OPTION
(a)
Discuss about ordinary point, singular point, regular singular point and irregular singular point for the differential equation x3(x−1)y′′+3(x−1)y′+7xy=0x^3 (x - 1)y'' + 3(x - 1)y' + 7xy = 0
3 Marks
(b)
Derive recurrence relation for the Bessel’s polynomial Jn(x)J_n(x).
4 Marks
(c)
Find the series solution of 2x(x−1)y′′−(x+1)y′+y=02x(x - 1)y'' - (x + 1)y' + y = 0 at x=0x = 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 - 2 (Maths 2) (Winter 2025, B.E. · Common Engineering, Sem 1). 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.

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