Differential Equations - Engineering Mathematics - Textile Engineering & Fibre Science Previous Year Questions

1

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2007

Using \exp(-x^2) as an integrating factor, the solution of the first order differential equation
y' - 2xy = 1
in terms of the error function [erf(x)] and a constant of integration c, is given by

A

\(y = [c - \frac{\sqrt{\pi}}{2} erf(x)] \exp(-x^2)\)

B

\(y = [\frac{\sqrt{\pi}}{2} erf(x) + c] \exp(-x^2)\)

C

\(y = [\frac{\sqrt{\pi}}{2} erf(x) + c] \exp(x^2)\)

D

\(y = [c - \frac{\sqrt{\pi}}{2} erf(x)] \exp(x^2)\)

Open complete paper

2

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2007

Given the second order differential equation
x^2 y'' + 2xy' - 6y = 0
which of the following constitutes its general solution

A

y = c_1 x^{-3} + c_2 x^2

B

y = c_1 x^3 + c_2 x^{-2}

C

y = c_1 \exp(3x) + c_2 \exp(-2x)

D

y = [c_1 + c_2 x] \exp(2x)

Open complete paper

3

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2007

The general solution of the third order ordinary differential equation
y''' - 3y' + 2y = 0
is given by

A

y = c_1 + c_2 \exp(-x) + c_3 \exp(2x)

B

y = [c_1 + c_2 x] \exp(-x) + c_3 \exp(2x)

C

y = c_1 + c_2 \exp(x) + c_3 \exp(-2x)

D

y = [c_1 + c_2 x] \exp(x) + c_3 \exp(-2x)

Open complete paper

4

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2007

By applying the method of separation of variables [u(x,t) = X(x)T(t)] to the heat equation
\[\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}\]
and assuming -k^2 as the separation constant, its solution is obtained as

A

u(x,t) = [c_1 \sin(kt) + c_2 \cos(kt)] \exp(-k^2 c^2 x)

B

u(x,t) = [c_1 \sinh(kt) + c_2 \cosh(kt)] \exp(-k^2 c^2 x)

C

u(x,t) = [c_1 \sin(kx) + c_2 \cos(kx)] \exp(-k^2 c^2 t)

D

u(x,t) = [c_1 \sinh(kx) + c_2 \cosh(kx)] \exp(-k^2 c^2 t)

Open complete paper

5

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2008

The second order differential equation \(x^2 \frac{d^3 y}{dx^3} + 5x \frac{dy}{dx} + 4y = 0\) under the transformation \(z = \ln x\), transforms to an ordinary differential equation with constant coefficients, which is given by

A

\(\frac{d^2 y}{dz^2} + 5 \frac{dy}{dz} + 4y = 0\)

B

\(\frac{d^2 y}{dz^2} + \frac{1}{5} \frac{dy}{dz} + 4y = 0\)

C

\(\frac{d^2 y}{dz^2} + 4 \frac{dy}{dz} + 4y = 0\)

D

\(\frac{d^2 y}{dz^2} + \frac{1}{4} \frac{dy}{dz} + 4y = 0\)

Open complete paper

6

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2008

The particular solution of the differential equation \(y'' + k^2 y = \alpha \sin \omega t\) where \(k \ne \omega\), is given by

A

\(\frac{\alpha \sin \omega t}{k^2 - \omega^2}\)

B

\(\frac{\alpha \sin \omega t}{k^2 + \omega^2}\)

C

\(\frac{\alpha \cos \omega t}{k^2 - \omega^2}\)

D

\(\frac{\alpha \cos \omega t}{k^2 + \omega^2}\)

Open complete paper

7

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2008

The Laplace transform of \(y(t)\) and its derivative are respectively defined as \(\int_0^\infty e^{-st} y(t) dt = Y(s)\) and \(\int_0^\infty e^{-st} y'(t) dt = s Y(s) - y(0)\). The Laplace transform of the initial value problem, \(y' - 2y = 0\), \(y(0) = 1\) gives

A

\(Y(s) = \frac{1}{s-2}\)

B

\(Y(s) = \frac{s}{s-2}\)

C

\(Y(s) = \frac{1}{s}\)

D

\(Y(s) = \frac{2}{s-2}\)

Open complete paper

8

2009 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2009

The general solution of the differential equation \(x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} - \frac{y}{x^2} = 0\) is

A

\(y = c_1 \cos x + c_2 \sin x\)

B

\(y = c_1 e^x + c_2 e^{-x}\)

C

\(y = \left( c_1 + \frac{c_2}{x} \right) e^{(1/x)}\)

D

\(y = (c_1 + c_2 x) e^x\)

Check answer

Open complete paper

9

2010 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2010

The order and degree of following differential equation are \[ \left[2+\left(\frac{dy}{dx}\right)^2\right]^2 = 5\frac{d^2y}{dx^2} \]

A

4 and 2 respectively

B

2 and 2 respectively

C

2 and 1 respectively

D

2 and 5 respectively

Open complete paper

10

2010 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2010

Solution of the differential equation \(\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 9x = 0\) will be

A

\(x = (c_1 + c_2t)e^{3t}\)

B

\(x = (c_1 + c_2t)e^{-3t}\)

C

\(x = c_1e^{3t} + c_2e^{-3t}\)

D

\(x = c_1e^{3t} + c_2e^{-3t}\)

Open complete paper

11

2011 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2011

The order and degree of the following differential equation are \((\frac{dy}{dx})^2 + \frac{dy}{dx} = y^2\)

A

order 1, degree 1

B

order 1, degree 2

C

order 2, degree 1

D

order 2, degree 2

Open complete paper

12

2013 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2013

The particular integral of \(\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 6y = e^{2x}\) is

A

\(e^{2x}/20\)

B

\(e^{2x}/12\)

C

\(2e^{2x}\)

D

\(4e^{2x}\)

Open complete paper

13

2014 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2014

A very large tank contains 10 liters of pure water. Salt solution (20 g/l) is pumped into the tank at 2 liter per minute. The salt concentration in g/l in the tank after a very long time will be

A

5

B

10

C

20

D

40

Open complete paper

14

2014 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2014

The integrating factor for solving the differential equation \[ \frac{dy}{dx} - \cos(x) + y\sin(x) = 2\cos^3(x) \] is

A

\sin(x)

B

\cos(x)

C

\tan(x)

D

\sec(x)

Open complete paper

15

2016 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2016

The following partial differential equation $U_{xx} + U_{yy} = 0$ is of the type

A

Elliptic

B

Parabolic

C

Hyperbolic

D

Mixed type

Open complete paper

16

2016 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2016

The integrating factor of \((2 \cos y + 4x^2)dx - x \sin y dy = 0\) is

A

\(-x\)

B

\(x\)

C

\(x^2\)

D

\(-x^2\)

Open complete paper

17

2017 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2017

If a solution curve of the differential equation \( x \frac{dy}{dx} = y + 2x^3 \) passes through the point (1,0), then this curve also passes through the point

A

(-1,0)

B

(0,-1)

C

(2,10)

D

(-2,6)

Open complete paper

18

2018 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2018

If \(y(x)\) is the solution of the differential equation \(y y' = 8x\), \(y(0) = 2\), then the absolute value of \(y(2)\) is ____________________

Check answer

Open complete paper

19

2019 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2019

One of the points which lies on the solution curve of the following differential equation \(2xy\,dx + (x^2 + y^2)\,dy = 0\) with the initial condition \(y(1) = 1\) is

A

(-1, 1)

B

(0, 0)

C

(0, 1)

D

(2, 1)

Open complete paper

20

2020 · Textile Engineering & Fibre Science · Engineering Mathematics · Differential Equations

Textile Engineering & Fibre Science (TF) 2020

The integrating factor of the differential equation \( \frac{dy}{dx} + y = e^{-x} \) is

A

\(e^x \)

B

\(e^{-x} \)

C

\(x e^{-x} \)

D

\(x e^x \)

Open complete paper