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Gujarat Technological UniversityWinter 2023 Examination

GTU 3110015 Mathematics - 2 (Maths 2) Winter 2023 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)
Define Solenoidal Vector field. Find the constant aa such that the vector (x+3)i^+(y−2z)j^+(x+az)k^(x + 3)\hat{i} + (y - 2z)\hat{j} + (x + az)\hat{k} is solenoidal.
3 Marks
(b)

If F⃗=3x2yi^+(x3−2yz2)j^+(3z2−2y2z)k^\vec{F} = 3x^2 y \hat{i} + (x^3 - 2yz^2)\hat{j} + (3z^2 - 2y^2 z)\hat{k} is conservative, then find:

iits scalar potential ϕ\phi.
iithe work done in moving from (2,1,1)(2, 1, 1) to (3,0,1)(3, 0, 1).
4 Marks
(c)
Verify Green’s theorem in the plane for ∮C(xy2−2xy) dx+(x2y+3) dy\oint_C (xy^2 - 2xy) \, dx + (x^2 y + 3) \, dy where CC is the region bounded by the rectangle with vertices (−1,0),(1,0),(1,1)(-1, 0), (1, 0), (1, 1) and (−1,1)(-1, 1).
7 Marks

Question 2

14 MarksMedium
(a)
State first shifting theorem for Laplace transform. Evaluate:
iL[e−tt5]\mathcal{L}[e^{-t} t^5]
iiL−1[1s2+2s+2]\mathcal{L}^{-1}\left[\frac{1}{s^2 + 2s + 2}\right]
3 Marks
(b)
Find:
iL[1−cos⁡2tt]\mathcal{L}\left[\frac{1 - \cos 2t}{t}\right]
iiL−1[54(s2+9)(s2−9)]\mathcal{L}^{-1}\left[\frac{54}{(s^2 + 9)(s^2 - 9)}\right]
4 Marks
(c)
Using Laplace transform solve the initial value problem:

y′′−y=t,with y(0)=1,  y′(0)=−1y'' - y = t, \quad \text{with } y(0) = 1, \; y'(0) = -1

7 Marks
OR OPTION
(c)
Find the Fourier sine integral of f(x)=e−bxf(x) = e^{-bx} and show that π2e−bx=∫0∞λsin⁡λxλ2+b2 dλ\frac{\pi}{2} e^{-bx} = \int_0^\infty \frac{\lambda \sin \lambda x}{\lambda^2 + b^2} \, d\lambda
7 Marks

Question 3

14 MarksMedium
(a)
Define linear differential equation. Solve: dydx+y=e−x\frac{dy}{dx} + y = e^{-x}, given that y(1)=1y(1) = 1.
3 Marks
(b)
Solve: (i) (D2+7D−18)y=0(D^2 + 7D - 18)y = 0, (ii) y′′+y′+2y=0y'' + y' + 2y = 0.
4 Marks
(c)
iSolve: (x+y−1) dx+(2x+2y−3) dy=0(x + y - 1) \, dx + (2x + 2y - 3) \, dy = 0
iiSolve: x2p2−4y2=0x^2 p^2 - 4y^2 = 0.
7 Marks
OR OPTION
(a)
Solve: y2dxdy+yx=y3x2y^2 \frac{dx}{dy} + yx = y^3 x^2.
3 Marks
(b)
Determine a second solution of x2d2ydx2−2y=0,x>0x^2 \frac{d^2 y}{dx^2} - 2y = 0, x > 0, where first solution is y1=1xy_1 = \frac{1}{x}.
4 Marks
(c)
iSolve: p=sin⁡(y−px)p = \sin(y - px)
iiSolve: (x3+y3) dx−xy2 dy=0(x^3 + y^3) \, dx - xy^2 \, dy = 0
7 Marks

Question 4

14 MarksMedium
(a)
Solve: (D2+3D+2)y=sin⁡2x(D^2 + 3D + 2)y = \sin 2x.
3 Marks
(b)
Solve: x3y′′′+x2y′′=x3x^3 y''' + x^2 y'' = x^3.
4 Marks
(c)
Using the method of undetermined coefficients solve y′′+9y=2x2y'' + 9y = 2x^2.
7 Marks
OR OPTION
(a)
Solve: (D2+4D)y=x+x2(D^2 + 4D)y = x + x^2.
3 Marks
(b)
Solve: (D2−6D+9)y=x2e3x(D^2 - 6D + 9)y = x^2 e^{3x}.
4 Marks
(c)
Using the method of variation of parameters solve: y′′−2y′+y=xexsin⁡xy'' - 2y' + y = x e^x \sin x.
7 Marks

Question 5

14 MarksMedium
(a)
Determine the singular points of the differential equation 2x(x−2)2y′′+3xy′+(x−2)y=02x(x - 2)^2 y'' + 3xy' + (x - 2)y = 0 and classify them as regular or irregular.
3 Marks
(b)
Find the series solution of y′′−y′=0y'' - y' = 0.
4 Marks
(c)
Solve the differential equation 3xy′′−(x−2)y′+2y=03xy'' - (x - 2)y' + 2y = 0 by using Frobenius method.
7 Marks
OR OPTION
(a)
Discuss about the ordinary points, regular singular points and irregular singular points for the differential equation x3y′′+5xy′+3y=0x^3 y'' + 5xy' + 3y = 0.
3 Marks
(b)
Find the power series solution of y′+2xy=0y' + 2xy = 0 in powers of xx.
4 Marks
(c)
iExpress J3(x)J_3(x) in terms of J1(x)J_1(x) and J2(x)J_2(x).
iiExpress f(x)=4x3+6x2+7x+1f(x) = 4x^3 + 6x^2 + 7x + 1 in terms of Legendre’s polynomials.
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 2023, 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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