Differential Equations - Engineering Mathematics - Production & Industrial Engineering Previous Year Questions

1

2008 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2008

For the partial differential equation \(\frac{\partial^2 u}{\partial x^2} = \pi^2 \frac{\partial u}{\partial t}\) in the domain 0≤x≤1 with boundary conditions u(0,t)=0 and u(1,t)=0 and initial condition u(x,0)=sin(πx), the solution of the differential equation is

A

e^{-t} sin(πx)

B

e^{t} sin(πx)

C

e^{-π²t} sin(πx)

D

e^{π²t} sin(πx)

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2

2008 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2008

The solutions of the differential equation \(\frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} + 2y = 0\) are

A

e^{-(1+i)x}, e^{-(1-i)x}

B

e^{(1+i)x}, e^{(1-i)x}

C

e^{-(1+i)x}, e^{(1+i)x}

D

e^{(1+i)x}, e^{-(1-i)x}

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3

2008 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2008

For the partial differential equation \(\frac{\partial^2 u}{\partial x^2} = \pi^2 \frac{\partial u}{\partial t}\) in the domain \(0 \leq x \leq 1\) with boundary conditions \(u(0, t) = 0\) and \(u(1, t) = 0\) and initial condition \(u(x, 0) = \sin(\pi x)\), the solution of the differential equation is

A

\(e^{-t} \sin(\pi x)\)

B

\(e^{t} \sin(\pi x)\)

C

\(e^{-\pi^2 t} \sin(\pi x)\)

D

\(e^{\pi^2 t} \sin(\pi x)\)

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4

2009 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2009

The homogeneous part of the differential equation \(\frac{d^2 y}{dx^2} + p\frac{dy}{dx} + qy = r\) \((p, q \text{ and } r \text{ are constants})\) has real distinct roots if

A

\(p^2 - 4q > 0\)

B

\(p^2 - 4q < 0\)

C

\(p^2 - 4q = 0\)

D

\(p^2 - 4q = r\)

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5

2009 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2009

The homogeneous part of the differential equation \(\frac{d^2y}{dx^2} + p\frac{dy}{dx} + qy = r\) \((p,q \text{ and } r \text{ are constants})\) has real distinct roots if

A

\(p^2 - 4q > 0\)

B

\(p^2 - 4q < 0\)

C

\(p^2 - 4q = 0\)

D

\(p^2 - 4q = r\)

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6

2009 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2009

The solution of the differential equation \(\frac{d^2 y}{dx^2} = 0\) with boundary conditions (i) \(\frac{dy}{dx} = 1\) at \(x = 0\) and (ii) \(\frac{dy}{dx} = 1\) at \(x = 1\), is

A

\(y = 1\)

B

\(y = x\)

C

\(y = x + C\), where C is an arbitrary constant.

D

\(y = C_1x + C_2\), where \(C_1\) and \(C_2\) are arbitrary constants.

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7

2010 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2010

Which one of the following differential equations has a solution given by the function \( y = 5\sin\left( 3x + \frac{\pi}{3} \right) \)?

A

\(\frac{dy}{dx} - \frac{5}{3}\cos(3x) = 0\)

B

\(\frac{dy}{dx} + \frac{5}{3}\cos(3x) = 0\)

C

\(\frac{d^2y}{dx^2} + 9y = 0\)

D

\(\frac{d^2y}{dx^2} - 9y = 0\)

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8

2010 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2010

The solution of the differential equation \[\frac{dy}{dx} - y^2 = 1\] satisfying the condition y(0) = 1 is

A

y = e^{2x}

B

\(y = \sqrt{x}\)

C

\(y = \cot\left(x + \frac{\pi}{4}\right)\)

D

\(y = \tan\left(x + \frac{\pi}{4}\right)\)

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9

2011 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2011

The solution of the differential equation \frac{d^2y}{dx^2} + 6\frac{dy}{dx} + 9y = 9x + 6 with C_1 and C_2 as constants is

A

y = (C_1x + C_2)e^{-3x}

B

y = C_1e^{3x} + C_2e^{-3x} + x

C

y = (C_1x + C_2)e^{-3x} + x

D

y = (C_1x + C_2)e^{3x} + x

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10

2011 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2011

The solution of the differential equation \( \frac{d^2 y}{dx^2} + 6\frac{dy}{dx} + 9y = 9x + 6 \) with \( C_1 \) and \( C_2 \) as constants is

A

\(y = (C_1 x + C_2) e^{-3x} \)

B

\(y = C_1 e^{3x} + C_2 e^{-3x} + x \)

C

\(y = (C_1 x + C_2) e^{-3x} + x \)

D

\(y = (C_1 x + C_2) e^{3x} + x \)

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11

2012 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2012

Consider the differential equation \(x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - 4y = 0\) with the boundary conditions of \(y(0) = 0\) and \(y(1) = 1\). The complete solution of the differential equation is

A

\(x^2\)

B

\(\sin\left(\frac{\pi x}{2}\right)\)

C

\(e^x \sin\left(\frac{\pi x}{2}\right)\)

D

\(e^{-x} \sin\left(\frac{\pi x}{2}\right)\)

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12

2013 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2013 [Session 1]

The partial differential equation \(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2}\) is a

A

Linear equation of order 2

B

Non-linear equation of order 1

C

Linear equation of order 1

D

Non-linear equation of order 2

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13

2013 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2013 [Session 1]

The solution to the differential equation \frac{d^2 u}{d x^2} - k \frac{d u}{d x} = 0 where k is a constant, subjected to the boundary conditions u(0) = 0 and u(L) = U, is

A

\(u = U \frac{x}{L}\)

B

\(u = U \left( \frac{1 - e^{kx}}{1 - e^{kL}} \right)\)

C

\(u = U \left( \frac{1 - e^{-kx}}{1 - e^{-kL}} \right)\)

D

\(u = U \left( \frac{1 + e^{kx}}{1 + e^{kL}} \right)\)

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14

2013 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2013 [Session 1]

The function f(t) satisfies the differential equation \frac{d^2 f}{dt^2} + f = 0 and the auxiliary conditions, f(0) = 0, \frac{df}{dt}(0) = 4. The Laplace transform of f(t) is given by

A

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

B

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

C

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

D

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

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15

2013 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2013 [Session 2]

The partial differential equation ∂u/∂t + u ∂u/∂x = ∂²u/∂x² is a

A

linear equation of order 2

B

non-linear equation of order 1

C

linear equation of order 1

D

non-linear equation of order 2

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16

2013 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2013 [Session 2]

The function \( f(t) \) satisfies the differential equation \( \frac{d^2 f}{dt^2} + f = 0 \) and the auxiliary conditions, \( f(0) = 0, \frac{df}{dt}(0) = 4 \). The Laplace transform of \( f(t) \) is given by

A

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

B

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

C

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

D

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

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17

2013 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2013 [Session 2]

The solution to the differential equation \( \frac{d^2 u}{dx^2} - k \frac{du}{dx} = 0 \) where \( k \) is a constant, subjected to the boundary conditions \( u(0) = 0 \) and \( u(L) = U \), is

A

\(u = U \frac{x}{L} \)

B

\(u = U \left( \frac{1 - e^{-kx}}{1 - e^{-kL}} \right) \)

C

\(u = U \left( \frac{1 - e^{-kx}}{1 - e^{kL}} \right) \)

D

\(u = U \left( \frac{1 + e^{kx}}{1 + e^{kL}} \right) \)

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18

2013 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2013 [Session 4]

The function \( f(t) \) satisfies the differential equation \( \frac{d^2 f}{dt^2} + f = 0 \) and the auxiliary conditions, \( f(0) = 0 \), \( \frac{df}{dt}(0) = 4 \). The Laplace transform of \( f(t) \) is given by

A

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

B

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

C

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

D

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

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19

2013 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2013 [Session 4]

The solution to the differential equation \(\frac{d^2u}{dx^2} - k\frac{du}{dx} = 0\) where \(k\) is a constant, subjected to the boundary conditions \(u(0) = 0\) and \(u(L) = U\), is

A

\(u = U\frac{x}{L}\)

B

\(u = U\left(\frac{1 - e^{kx}}{1 - e^{kL}}\right)\)

C

\(u = U\left(\frac{1 - e^{-kx}}{1 - e^{-kL}}\right)\)

D

\(u = U\left(\frac{1 + e^{kx}}{1 + e^{kL}}\right)\)

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20

2018 · Production & Industrial Engineering · Engineering Mathematics · Differential Equations

Production & Industrial Engineering (PI) 2018

Consider the differential equation \(2\frac{d^2y}{dt^2} + 8y = 0\) with initial conditions: at \(t=0\), \(y=0\) and \(\frac{dy}{dt}=10\). The value of \(y\) (up to two decimal places) at \(t=1\) is ______

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