JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year ORDINARY DIFFERENTIAL EQUATIONS AND VECTOR CALCULUS Previous Question Paper (January/February - 2024 - R22)

B.Tech | I Year | II Semester | R22 | January/February - 2024

Answer PaperPaper Code: 182ARMaximum Marks: 60Time: 3 Hours

Note: This question paper contains two parts A and B. Part-A is a compulsory question. Part-B consists of ten questions (numbered from 2 to 11) carrying 10 marks each. From each unit, there are two questions and the student should answer one of them. Hence, the student should answer five questions from Part-B.

Main

1
Part-A: Compulsory sub-questions
10 marks
a
Write down the general form of a linear differential equation of first order.
1 marks

Answer

The general form of a first-order linear differential equation is

dy/dx + P(x)y = Q(x)
, where P and Q are functions of x only or constants.

b
Give an example of a family of curves which is self-orthogonal.
1 marks

Answer

A classic example of a self-orthogonal family of curves is the family of rectangular hyperbolas given by

xy = c
, where c is a constant.

c
Define the Legendre’s homogeneous differential equation.
1 marks

Answer

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d
Find the particular integral of
d^3y/dx^3 + 4dy/dx = sin 2x
.
1 marks

Answer

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e
Show that the Laplace transform of the unit step function u(t-a) is
e^{-as}/s
.
1 marks

Answer

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f
State the Convolution Theorem for Laplace transform.
1 marks

Answer

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g
Define a vector point function give an example.
1 marks

Answer

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h
Find the curl of
V = e^{xyz}(i + j + k)
at the point (1, 2, 3).
1 marks

Answer

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i
State Stoke’s Theorem.
1 marks

Answer

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j
If S is a closed surface and V is the volume of the region bounded by S then what is the value of
∯_S r · N dS
where
r = xi + yj + zk
.
1 marks

Answer

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210 marks
a
A bacterial population B is known to have a rate of growth directly proportional to B itself. If between noon and 2 PM the population triples, at what time, no controls being exerted, should B become 100 times what it was at noon.
5 marks

Answer

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b
Solve
x(dx/dy + y) = 1 - y
.
5 marks

Answer

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OR
310 marks
a
Solve
[1 + log(xy)]dx + [1 + x/y] dy = 0
.
5 marks

Answer

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b
Find the particular member of the orthogonal trajectories of
x^2 + cy^2 = 1
passing through the point (2, 1).
5 marks

Answer

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410 marks
a
Solve
(2x + 5)^2 d^2y/dx^2 - 6(2x + 5)dy/dx + 8y = 6x
.
5 marks

Answer

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b
Solve
d^2y/dx^2 - y = x sin x + x^2 e^x
.
5 marks

Answer

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OR
510 marks
a
Apply the method of variation of parameters to solve
d^2y/dx^2 + y = tan x
.
5 marks

Answer

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b
Solve
d^2y/dx^2 - 2dy/dx + 5y = 0
, y(0) = -3 and dy/dx = 1 at x = 0.
5 marks

Answer

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610 marks
a
Find the Laplace transform of the Half wave rectifier given as
f(t) = { sin ωt, 0 < t < π/ω; 0, π/ω < t < 2π/ω }
.
5 marks

Answer

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b
Solve by using the Laplace transform
y'' + 2y' + 5y = e^{-t} sin t
, y(0) = 0, and y'(0) = 1.
5 marks

Answer

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OR
710 marks
a
Evaluate
L^{-1}{ (e^{-s} - 3e^{-3s})/s^2 }
.
5 marks

Answer

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b
Evaluate
∫_0^∞ e^{-t} ((cos at - cos bt)/t) dt
with the help of Laplace transform.
5 marks

Answer

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810 marks
a
If
v = (xi + yj + zk) / √(x^2 + y^2 + z^2)
, find the value of divergence (v).
5 marks

Answer

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b
Find the value of n for which the vector
r^n r
is solenoidal, where
r = xi + yj + zk
.
5 marks

Answer

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OR
910 marks
a
Find the divergence and curl of
v = (xyz)i + (3x^2y)j + (xz^2 - y^2z)k
at (2, -1, 1).
5 marks

Answer

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b
Prove that
(y^2 - z^2 + 3yz - 2x)i + (3xz + 2xy)j + (3xy - 2xz + 2z)k
is both solenoidal and irrotational.
5 marks

Answer

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1010 marks
a
A vector field is given by
F = (sin y)i + x(1 + cos y)j
. Evaluate the line integral over a circular path
x^2 + y^2 = a^2, z = 0
.
5 marks

Answer

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b
Evaluate
∯_S (yz i + zx j + xy k) dS
over S, where S is the surface of the sphere
x^2 + y^2 + z^2 = a^2
in the first octant.
5 marks

Answer

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OR
1110 marks
a
Use Green’s Theorem, evaluate
∫_C (x^2y dx + x^2 dy)
over C, where C is the boundary described counter clockwise of the triangle with vertices (0, 0), (1, 0), (1, 1).
5 marks

Answer

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b
Apply Stoke’s Theorem to find the value of
∫_C (y dx + z dy + x dz)
over C, where C is the curve of intersection of
x^2 + y^2 + z^2 = a^2
and
x + z = a
.
5 marks

Answer

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About this paper

This page contains the question paper of ORDINARY DIFFERENTIAL EQUATIONS AND VECTOR CALCULUS from JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD for B.Tech MNE-MIE I Year II Semester, January/February - 2024, R22.

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