Differential Equations - Engineering Mathematics - Aerospace Engineering Previous Year Questions

1

2007 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2007

A spring-mass-damper system with a mass of 1 kg is found to have a damping ratio of 0.2 and a natural frequency of 5 rad/s. The damping of the system is given by

A

2 Ns/m

B

2 N/s

C

0.2 kg/s

D

0.2 N/s

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2

2007 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2007

A 1 kg mass attached to a spring elongates it by 16mm. The mass is then pulled from its equilibrium position by 10mm and released from rest. Assuming the acceleration due to gravity of 9.81 m/s^2, the response of the mass in mm is given by

A

x = 10 sin 24.76t

B

x = 10 cos 24.76t

C

x = sin 16t

D

x = 10 cos 16t

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3

2007 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2007

Let a dynamical system be described by the differential equation \( 2 \frac{dx}{dt} + \cos x = 0 \). Which of the following differential equations describes this system in a close approximation sense for small perturbation about \( x = \pi / 4 \)?

A

\(2 \frac{dx}{dt} + \sin x = 0 \)

B

\(2 \frac{dx}{dt} - \frac{1}{\sqrt{2}} x = 0 \)

C

\(\frac{dx}{dt} + \cos x = 0 \)

D

\(\frac{dx}{dt} + x = 0 \)

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4

2007 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2007

The inverse Laplace transform of \( F(s) \) is

A

\(2e^{-2t} + 20e^{-20t} \)

B

\(\frac{4}{9} e^{-2t} + \frac{5}{9} e^{-20t} \)

C

\(5e^{-2t} + 2e^{-20t} \)

D

\(\frac{4}{9} e^{-2t} + \frac{9}{5} e^{-20t} \)

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5

2008 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2008

In a spring-mass-damper single degree of freedom system, the mass is 2 kg and the undamped natural frequency is 20 Hz. The critical damping constant of the system is

A

160\(\pi\) N.s/m

B

80\(\pi\) N.s/m

C

1 N.s/m

D

0 N.s/m

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6

2008 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2008

Which of the following is a solution of \(\frac{d^2 y}{dx^2} + 2\frac{dy}{dx} + y = 0\)?

A

\(e^{-x} + xe^{-x}\)

B

\(e^x + xe^{-x}\)

C

\(e^x + e^{-x}\)

D

\(e^{-x} + xe^x\)

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7

2008 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2008

Suppose the non-constant functions \(F(x)\) and \(G(t)\) satisfy \(\frac{d^2 F}{dx^2} + p^2 F = 0\), \(\frac{dG}{dt} + c^2 p^2 G = 0\), where \(p\) and \(c\) are constants. Then the function \(u(x,t) = F(x)G(t)\) definitely satisfies

A

\(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\)

B

\(\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}\)

C

\(\nabla^2 u = 0\)

D

\(\frac{\partial^2 u}{\partial t^2} + c^2 u^2 = 0\)

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8

2008 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2008

Let \(Y(s)\) denote the Laplace transform \(L[y(t)]\) of the function \(y(t) = \cosh(at) \sin(at)\). Then

A

\(L\left(\frac{dy}{dt}\right) = \frac{dY}{ds}, \; L(t y(t)) = sY(s)\)

B

\(L\left(\frac{dy}{dt}\right) = sY(s), \; L(t y(t)) = -\frac{dY}{ds}\)

C

\(L\left(\frac{dy}{dt}\right) = \frac{dY}{ds}, \; L(t y(t)) = Y(s-1)\)

D

\(L\left(\frac{dy}{dt}\right) = sY(s), \; L(t y(t)) = e^{-s} Y(s)\)

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9

2009 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2009

The ordinary differential equation \(\frac{d^2y}{dx^2} + ky = 0\) where \(k\) is real and positive

A

is non-linear

B

has a characteristic equation with one real and one complex root

C

has a characteristic equation with two real roots

D

has a complementary function that is simple harmonic

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10

2009 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2009

For a critically damped single degree of freedom spring – mass – damper system with a damping constant \(c\) of 4 Ns/m and spring constant \(k\) of 16 N/m, the system mass \(m\) is

A

0.5 kg

B

0.25 kg

C

2 kg

D

4 kg

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11

2009 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2009

For the spring-mass system shown below, the natural frequencies are

A

0 and \(\sqrt{\frac{k(m_1+m_2)}{m_1m_2}}\)

B

0 and \(\sqrt{\frac{k(m_1+m_2)}{2m_1m_2}}\)

C

0 and \(\sqrt{\frac{k}{m_1+m_2}}\)

D

0 and \(\sqrt{\frac{k}{2(m_1+m_2)}}\)

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12

2009 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2009

The inverse Laplace transform of \(F(s) = \frac{(s+1)}{(s+4)(s-3)}\) is

A

\(\frac{3}{7}e^{4t} + \frac{4}{7}e^{-3t}\)

B

\(\frac{3}{7}e^{-4t} + \frac{4}{7}e^{3t}\)

C

\(\frac{5}{7}e^{-4t} + \frac{6}{7}e^{3t}\)

D

\(\frac{5}{7}e^{4t} + \frac{6}{7}e^{-3t}\)

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13

2010 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2010

The linear second order partial differential equation \(3\frac{\partial^2 \phi}{\partial x^2} + 3\frac{\partial^2 \phi}{\partial x \partial y} + 2\frac{\partial^2 \phi}{\partial y^2} + 9 = 0\) is

A

Parabolic

B

Hyperbolic

C

Elliptic

D

None of the above

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14

2010 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2010

The concentration x of a certain chemical species at time t in a chemical reaction is described by the differential equation \(\frac{dx}{dt} + kx = 0\), with \(x(t=0) = x_0\). Given that e is the base of the natural logarithms, the concentration x at \(t = \frac{1}{k}\) is

A

falls to the value \(0.5x_0\)

B

rises to the value \(2x_0\)

C

falls to the value \(\frac{x_0}{e}\)

D

rises to the value \(ex_0\)

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15

2010 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2010

Given that the Laplace transform of \(y(t) = e^{-t} (2 \cos 2t - \sin 2t)\) is \(Y(s) = \frac{2s}{(s+1)^2 + 4}\), the Laplace transform of \(y_1(t) = e^{t} (2 \cos 2t - \sin 2t)\) is

A

\(\frac{2(s-2)}{(s-1)^2 + 4}\)

B

\(\frac{2(s+2)}{(s+3)^2 + 4}\)

C

\(\frac{2(s+2)}{(s+1)^2 + 4}\)

D

\(\frac{2(s-1)}{(s-1)^2 + 4}\)

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16

2011 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2011

The solution of \( \frac{dy}{dt} = y^3 e^t t^2 \) with initial condition \( y(0) = 1 \) is given by

A

\(\frac{1}{9} e^t (t+3)^2 \)

B

\(\sqrt{5 + 2e^t (t^2 - 2t + 2)} \)

C

\(\frac{4e^t}{(t+2)^2} \)

D

\(\sqrt{5 - 2e^t (t^2 - 2t + 2)} \)

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17

2011 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2011

To solve the PDE, the number of boundary conditions (BC) and initial conditions (IC) needed are

A

4 BC, 3 IC

B

2 BC, 2 IC

C

2 BC, 4 IC

D

4 BC, 2 IC

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18

2012 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2012

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

A

\(Ae^{-t} + Be^{2t}\)

B

\(Ae^{-2t} + Be^{-t}\)

C

\(Ae^{-2t} + Be^{t}\)

D

\(Ae^t + Be^{2t}\)

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19

2012 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2012

The value of \(k\) for which the system of equations \(x + 2y + kz = 1; \quad 2x + ky + 8z = 3\) has no solution is

A

0

B

2

C

4

D

8

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20

2012 · Aerospace Engineering · Engineering Mathematics · Differential Equations

Aerospace Engineering (AE) 2012

If \(u(t)\) is a unit step function, the solution of the differential equation \(m \frac{d^2 x}{dt^2} + kx = u(t)\) in Laplace domain is

A

\(\frac{1}{s(ms^2 + k)}\)

B

\(\frac{1}{ms^2 + k}\)

C

\(\frac{s}{ms^2 + k}\)

D

\(\frac{1}{s^2(ms^2 + k)}\)

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