UnFoldPrep
Browse Archive
Gujarat Technological UniversitySummer 2025 Examination

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

B.E. · Common Engineering · Semester 1 · Subject Code: 3110015
Download Official GTU PDF
Share:
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
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Show that F⃗=(y2−z2+3yz−2x)i^+(3xz+2xy)j^+(3xy−2xz+2z)k^\vec{F} = (y^2 - z^2 + 3yz - 2x)\hat{i} + (3xz + 2xy)\hat{j} + (3xy - 2xz + 2z)\hat{k} is both solenoidal and irrotational.
3 Marks
(b)
Find Fourier integral representation of function f(x)={1∣x∣<10∣x∣>1f(x) = \begin{cases} 1 & |x| < 1 \\ 0 & |x| > 1 \end{cases} Hence, evaluate ∫0∞sin⁡ωcos⁡ωxω dω\int_0^\infty \frac{\sin \omega \cos \omega x}{\omega} \, d\omega
4 Marks
(c)
Verify Green’s theorem in the plane for ∫C[(3x2−8y2) dx+(4y−6xy) dy]\int_C \left[ (3x^2 - 8y^2) \, dx + (4y - 6xy) \, dy \right] where CC is the boundary of the region bounded by x=0,y=0,x+y=1x = 0, y = 0, x + y = 1.
7 Marks

Question 2

14 MarksMedium
(a)
Solve yex dx+(2y+ex) dy=0y e^x \, dx + (2y + e^x) \, dy = 0.
3 Marks
(b)
Find the directional derivative of ϕ=xy2+yz2\phi = x y^2 + y z^2 at the point (2,−1,1)(2, -1, 1) in the direction of the vector i^+2j^+2k^\hat{i} + 2\hat{j} + 2\hat{k}.
4 Marks
(c)
Find the power series solution of d2ydx2+xy=0\frac{d^2 y}{dx^2} + x y = 0
7 Marks
OR OPTION
(c)
Find the series solution of (1−x2)y′′−2xy′+2y=0(1 - x^2) y'' - 2x y' + 2y = 0.
7 Marks

Question 3

14 MarksMedium
(a)
Solve (4D2−4D+1)y=ex2(4D^2 - 4D + 1)y = e^{\frac{x}{2}}.
3 Marks
(b)
Solve (D2+6D+8)y=cos⁡2x(D^2 + 6D + 8)y = \cos^2 x.
4 Marks
(c)
Solve (D2−2D+3)y=x3+sin⁡x(D^2 - 2D + 3)y = x^3 + \sin x by using the method of undetermined coefficients.
7 Marks
OR OPTION
(a)
Solve (D2+a2)y=cos⁡ax(D^2 + a^2)y = \cos ax.
3 Marks
(b)
Solve d2ydx2+2dydx+2y=exsin⁡x+7\frac{d^2 y}{dx^2} + 2\frac{dy}{dx} + 2y = e^x \sin x + 7
4 Marks
(c)
Solve x2d2ydx2+xdydx+y=log⁡x⋅sin⁡(log⁡x)x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + y = \log x \cdot \sin(\log x) by Cauchy’s linear equation.
7 Marks

Question 4

14 MarksMedium
(a)
Find the Laplace Transform of f(t)=sinh⁡atf(t) = \sinh at.
3 Marks
(b)
Find the Laplace Transform of te4tcos⁡2tt e^{4t} \cos 2t.
4 Marks
(c)
Solve the differential equation using initial value problem: y′′+6y′=1,y(0)=2,  y′(0)=0y'' + 6y' = 1, \quad y(0) = 2, \; y'(0) = 0
7 Marks
OR OPTION
(a)
Find the inverse Laplace transform of log⁡s2+b2s2+a2\log \frac{s^2 + b^2}{s^2 + a^2}
3 Marks
(b)
Find the Laplace Transform of e−2tsin⁡2tcosh⁡tt\frac{e^{-2t} \sin 2t \cosh t}{t}
4 Marks
(c)
Find the Inverse Laplace Transform of 5s+3(s−1)(s2+2s+5)\frac{5s + 3}{(s - 1)(s^2 + 2s + 5)}
7 Marks

Question 5

14 MarksMedium
(a)
Solve dydx+ycot⁡x=2cos⁡x\frac{dy}{dx} + y \cot x = 2 \cos x
3 Marks
(b)
Solve 3x4p2−xp−y=03x^4 p^2 - xp - y = 0.
4 Marks
(c)
Using the method variation of parameter solve the differential equation (D2+1)y=xsin⁡x(D^2 + 1)y = x \sin x.
7 Marks
OR OPTION
(a)
Solve (xy−2y2) dx−(x2−3xy) dy=0(xy - 2y^2) \, dx - (x^2 - 3xy) \, dy = 0.
3 Marks
(b)
Solve dydx+1xy=y2x2\frac{dy}{dx} + \frac{1}{x} y = \frac{y^2}{x^2}
4 Marks
(c)
Solve (D4−16)y=e2x+x4(D^4 - 16)y = e^{2x} + x^4.
7 Marks
College Exam Groups

Studying for Mathematics - 2?

Circulate this solved paper with KaTeX formulas and 1-click AI step solvers to your batchmates on WhatsApp or Telegram.

Related papers

About this Examination Paper & Attribution

Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Mathematics - 2 (Maths 2) (Summer 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.

Download PDF