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JNTU B.Tech II MATHEMATICS-III Previous Question Paper (April - 2018 - R15)

B.Tech | II | II-II | R15 | April - 2018

Answer PaperPaper Code: 124DEMaximum Marks: 75Time: 3 Hours

Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit.

Part A

1.a

Determine the nature of the point x = 0 for the equation

(x2+1)y'' + (x2-1)y' + 2y = 0

2 marks

Answer

To determine the nature of the point x = 0, rewrite the differential equation in the standard form

y'' + P(x)y' + Q(x)y = 0
where
P(x) = (x2-1)/(x2+1)
and
Q(x) = 2/(x2+1)
. Evaluating these at
x = 0
, we find
P(0) = -1
and
Q(0) = 2
. Since both
P(x)
and
Q(x)
are analytic at
x = 0
, the point is an ordinary point.

1.b

Find the particular integral of

x2y'' - xy' + y = x

3 marks

Answer

The given equation is a Cauchy-Euler equation:

x2y'' - xy' + y = x
. To solve this, substitute x = ez, which implies z = ln(x). Let D = d/dz. The equation transforms into a linear differential equation with constant coefficients:

(D(D-1) - D + 1)y = ez
(D2 - 2D + 1)y = ez

The characteristic equation is (m - 1)2 = 0, giving roots m = 1, 1. The particular integral (PI) is given by:

PI = [1 / (D - 1)2] ez

Since the operator results in zero when applied to ez, we use the formula [1 / f(D)] eaz = [zk / fk(a)] eaz, where k is the multiplicity of the root:

PI = (z2 / 2!) ez

Substituting z = ln(x) and ez = x back into the expression, the particular integral is:

PI = (x(ln(x))2) / 2
1.c

Write the value of

J1/2(x)

2 marks

Answer

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1.d

Express

2 - 3x + 4x2
in terms of Legendre polynomial.

3 marks

Answer

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1.e

If zn, z0 are the values of z at the end points of the curve C and f(z) = 1 then find the value of

∫C f(z)dz

2 marks

Answer

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1.f

Show that

xy2
cannot be real part of an analytic function.

3 marks

Answer

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1.g

Evaluate

∫C (ez)/(z+1)2 dz
, where C is the circle |z-3| = 3.

2 marks

Answer

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1.h

Find the kind of singularity for the function

1/(sin z - cos z)

3 marks

Answer

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1.i

Find the fixed points of

w = (z-1)/(z+1)

2 marks

Answer

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1.j

Distinguish between conformal and isogonal mapping.

3 marks

Answer

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Part B

210 marks
a

Obtain the series solution of the equation y'' + xy = 0.

5 marks

Answer

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b

Find the series solution of 4xy'' + 2y' + y = 0.

5 marks

Answer

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OR
310 marks
a

Solve

d2y/dx2 - (1/x)dy/dx + y/x = log x/x2

5 marks

Answer

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b

Solve the equation 3x(d2y/dx2) + (1-x)dy/dx - y = 0 in power series.

5 marks

Answer

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410 marks
a

Prove that

∫-11 xPn(x)Pn-1(x) dx = 2n/(4n2-1)

5 marks

Answer

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b

Show that

J02 + 2J12 + J22 + J32 + ... = 1

5 marks

Answer

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OR
510 marks
a

If m1, m2 are roots of Jn(x) = 0, then prove that

∫01 xJn(m1x)Jn(m2x) dx = 0

5 marks

Answer

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b

Show that P0(x) = 1, P1(x) = x, P2(x) = (3x2-1)/2 and hence express 2x2 – 4x + 2 as a Legendre polynomial.

5 marks

Answer

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6

State and prove Cauchy’s Integral formula.

10 marks

Answer

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OR
7

Evaluate

∫c (y - x - 3x2i) dz
, where c consists of the line segments from z = 0 to z = i and the other from z = i to z = 1+i.

10 marks

Answer

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8

Evaluate

∫-∞∞ (z2 - z + 2)/(z4 + 10z2 + 9) dz

10 marks

Answer

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OR
9

Find Laurent expansion of

1/(z2 - 4z + 3)
for 1 < |z| < 3.

10 marks

Answer

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10

Find the bilinear transform which maps the points z = 0, -i, -1 into the points w = i, 1, 0. Find the image of the line y = mx under this transformation.

10 marks

Answer

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OR
11

Show that every bilinear transformation maps the circles in the z – plane onto the circles in the w – plane.

10 marks

Answer

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This page contains the question paper of MATHEMATICS-III from JNTU for B.Tech MME-MMT II II-II, April - 2018, R15.

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