Jawaharlal Nehru Technological University Hyderabad

Jawaharlal Nehru Technological University Hyderabad B.Tech II Year MATHEMATICS-III Previous Question Paper (September/October - 2023 - R15)

B.Tech | II Year | I Semester | R15 | September/October - 2023

Answer PaperPaper Code: 123AHMaximum Marks: 75Time: 3 Hours

Note: i) Question paper consists of Part A, Part B.

ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions.

iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.

Part A

1.a

Solve

x^2 (d^2y/dx^2) + y = 0

2 marks

Answer

The given equation is a Cauchy-Euler equation of the form

x²y'' + y = 0

. Substituting

x = e^z

z = ln(x)

, the equation transforms into a linear differential equation with constant coefficients:

D(D-1)y + y = 0

, where

D = d/dz

. This simplifies to

(D² - D + 1)y = 0

. The auxiliary equation is

m² - m + 1 = 0

, which has roots

m = (1 ± i√3)/2

. Thus, the general solution is

y = x^1/2 [C₁ cos(√3/2 ln x) + C₂ sin(√3/2 ln x)]

1.b

Locate and classify the ordinary points, regular singular point and irregular singular point of the differential equation:

x(x-1)^2(d^2y/dx^2) + 2(x-1)(dy/dx) + 4x^2y = 0

3 marks

Answer

To classify the points, we first write the differential equation in the standard form y'' + P(x)y' + Q(x)y = 0 by dividing by x(x-1)²:

y'' + [2 / (x(x-1))]y' + [4x / (x-1)²]y = 0

Here, P(x) = 2 / (x(x-1)) and Q(x) = 4x / (x-1)².

Ordinary points: All points x ≠ 0 and x ≠ 1 are ordinary points because P(x) and Q(x) are analytic at these values.

Singular points: The points x = 0 and x = 1 are singular points.

At x = 0: Calculate lim_x→0 xP(x) = lim_x→0 2/(x-1) = -2 and lim_x→0 x²Q(x) = lim_x→0 4x³/(x-1)² = 0. Since both limits are finite, x = 0 is a regular singular point.
At x = 1: Calculate lim_x→1 (x-1)P(x) = lim_x→1 2/x = 2 and lim_x→1 (x-1)²Q(x) = lim_x→1 4x = 4. Since both limits are finite, x = 1 is a regular singular point.

1.c

State Legendre’s differential equation and write its general solution.

2 marks

Answer

1.d

Prove or disprove that

\int_0^1 (J_0(x))^2 dx = (J_0(0))^2

3 marks

Answer

1.e

Show that

f(z) = Im(z)

is not differentiable anywhere.

2 marks

Answer

1.f

Determine

\int_C Re(z) dz

where C is the shortest path from 1+i to 3+2i.

3 marks

Answer

1.g

Find the singular point(s) of the function

f(z) = (z-sin(z))/z^3

and also write the nature of these singular point(s).

2 marks

Answer

1.h

Compute the residue of

f(z) = z^3 / (z-1)^2

at its poles.

3 marks

Answer

1.i

Define cross-ratio of four points.

2 marks

Answer

1.j

Evaluate the critical and invariant points of (i)

f(z) = z^2

, (ii)

f(z) = z

, if any.

3 marks

Answer

Part B

210 marks

State Cauchy’s homogeneous linear equation.

2 marks

Answer

Solve

x^2(d^2y/dx^2) - x(dy/dx) + 3y = sin(ln x)

8 marks

Answer

Using power series method solve the initial value problem

x^2(d^2y/dx^2) + 2y = 0, y(1) = 2, y'(1) = 4

10 marks

Answer

410 marks

Prove or disprove

d/dx [x^n J_n(x)] = x^n J_{n-1}(x)

4 marks

Answer

Prove

J_0^2 + 2(J_1^2 + J_2^2 + ...) = 1

using Jacobi series.

6 marks

Answer

510 marks

Show that

\sum_{n=0}^{\infty} P_n(x) t^n = (1-2xt+t^2)^{-1/2}

6 marks

Answer

Express

x^3 + 3x^2 + 4x + 3

in terms of Legendre polynomial.

4 marks

Answer

610 marks

Show that

f(z) = (x^3(1+i) - y^3(1-i)) / (x^2+y^2)

for

z \neq 0

and

f(0) = 0

satisfies the Cauchy-Riemann equations at origin but derivative of f(z) at origin does not exist.

7 marks

Answer

Determine constant b such that

u = e^{bx} cos(5y)

is harmonic function.

3 marks

Answer

Determine F(2), F(i-3), F'(i) and F''(i-2) if

F(z) = \int_c (5z^2-4z+3)/(z-\alpha) dz

, where c is the ellipse

16x^2 + 9y^2 = 144

10 marks

Answer

Expand

f(z) = 1/((z+1)(z+2))

in Taylor's series or Laurent's series valid for (a) |z| < 1, (b) |z| > 1, and (c) 0 < |z+1| < 2.

10 marks

Answer

910 marks

State residue theorem.

2 marks

Answer

Prove that

\int_0^{\infty} 1/(x^6+1) dx = \pi/3

, using the residue theorem.

8 marks

Answer

Determine the linear fractional transformation that sends the points z = 0, i, 2i into the points w = 5, \infty, i/3 respectively. Hence find a) The image of |z| < 1 under this transformation. b) The critical and invariant points of this transformation.

10 marks

Answer

1110 marks

What is the meant by conformal mapping. Write the sufficient conditions for w = f(z) to be conformal.

3 marks

Answer

Discuss the transformation w = z + 1/z in detail.

7 marks

Jawaharlal Nehru Technological University Hyderabad B.Tech II Year MATHEMATICS-III Previous Question Paper (September/October - 2023 - R15)

Part A

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Part B

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