Differential Equations - Engineering Mathematics - Chemical Engineering Previous Year Questions

1

2007 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2007

The initial condition for which the following equation
\((x^2 + 2x) \frac{dy}{dx} = 2(x+1)y; \quad y(x_0) = y_0\)
has infinitely many solutions, is

A

\(y(x=0)=5\)

B

\(y(x=0)=1\)

C

\(y(x=2)=1\)

D

\(y(x=-2)=0\)

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2

2007 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2007

Given that the Laplace transform of the function below over a single period \(0 < t < 2\) is \(\frac{1}{s^2}(1-e^{-s})^2\), the Laplace transform of the periodic function over \(0 < t < \infty\) is

A

\(\frac{1}{s}(1-e^{-s})^2\)

B

\(\frac{1}{s}(1-e^{-s})^2\)

C

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

D

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

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3

2007 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2007

The solution of the following differential equation
\( x \frac{dy}{dx} + y(x^2 - 1) = 2x^3 \)
is

A

0

B

\(2 + ce^{\frac{x^2}{2}} \)

C

\(c_1 x + c_2 x^2 \)

D

\(2x + cxe^{-\frac{x^2}{2}} \)

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4

2007 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2007

The family of curves that is orthogonal to
\( xy = c \)
is

A

\(y = c_1 x \)

B

\(y = c_1 / x \)

C

\(y^2 + x^2 = c_1 \)

D

\(y^2 - x^2 = c_1 \)

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5

2008 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2008

Q.1 Which ONE of the following is NOT an integrating factor for the differential equation \( xdy - ydx = 0 \)?

A

\(\frac{1}{x^2} \)

B

\(\frac{1}{y^2} \)

C

\(\frac{1}{xy} \)

D

\(\frac{1}{x+y} \)

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6

2008 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2008

Q.2 Which ONE of the following is NOT a solution of the differential equation \( \frac{d^2 y}{dx^2} + y = 1 \)?

A

\(y = 1 \)

B

\(y = 1 + \cos x \)

C

\(y = 1 + \sin x \)

D

\(y = 2 + \sin x + \cos x \)

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7

2008 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2008

Which ONE of the following transformations \( u = f(y) \) reduces \( \frac{dy}{dx} + Ay^3 + By = 0 \) to a linear differential equation? (A and B are positive constants)

A

\(u = y^{-3} \)

B

\(u = y^{-2} \)

C

\(u = y^{-1} \)

D

\(u = y^2 \)

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8

2008 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2008

The Laplace transform of the function \( f(t) = t \sin t \) is

A

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

B

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

C

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

D

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

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9

2009 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2009

The general solution of the differential equation \(\frac{d^2y}{dx^2} - \frac{dy}{dx} - 6y = 0\), with \(C_1\) and \(C_2\) as constants of integration, is

A

\(C_1 e^{-3x} + C_2 e^{-2x}\)

B

\(C_1 e^{3x} + C_2 e^{-2x}\)

C

\(C_1 e^{3x} + C_2 e^{2x}\)

D

\(C_1 e^{-3x} + C_2 e^{2x}\)

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10

2009 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2009

The inverse Laplace transform of \(\frac{1}{{2{s^2} + 3s + 1}}\) is

A

\(e^{-t/2} - e^{-t}\)

B

\(2e^{-t/2} - e^{-t}\)

C

\(e^{-t} - 2e^{-t/2}\)

D

\(e^{-t} - e^{-t/2}\)

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11

2010 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2010

The solution of the differential equation \[ \frac{d^2 y}{dt^2} + 2 \frac{dy}{dt} + 2y = 0 \] with the initial conditions \( y(0) = 0, \; \frac{dy}{dt}\big|_{t=0} = -1 \), is

A

-t \sin t

B

-e^{-t} (1 - \cos t)

C

-(t + \sin t) / 2

D

-e^{-t} \sin t

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12

2011 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2011

Which ONE of the following functions \(y(x)\) has the slope of its tangent equal to \(\frac{a x}{y}\)?
Note: a and b are real constants

A

\(y = \frac{x+b}{a}\)

B

\(y = ax + b\)

C

\(y = \sqrt{\frac{x^2 + b}{a}}\)

D

\(y = \sqrt{ax^2 + b}\)

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13

2011 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2011

Which ONE of the following choices is a solution of the differential equation given below? \(\frac{dy}{dx} = \frac{y^2}{x} + \frac{y}{x} - \frac{2}{x}\) Note: \(c\) is a real constant

A

\(y = \frac{c - x^2}{c + 2x^2}\)

B

\(y = \frac{c + 2x^2}{c - x^2}\)

C

\(y = \frac{c - x^3}{c + 2x^3}\)

D

\(y = \frac{c + 2x^3}{c - x^3}\)

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14

2012 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2012

If \( a \) and \( b \) are arbitrary constants, then the solution to the ordinary differential equation
\[ \frac{d^2 y}{dx^2} - 4y = 0 \]
is

A

\(y = ax + b \)

B

\(y = ae^{-x} \)

C

\(y = a\sin 2x + b\cos 2x \)

D

\(y = a\cosh 2x + b\sinh 2x \)

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15

2013 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2013

The solution of the differential equation \(\frac{dy}{dx} - y^2 = 0\), given \(y = 1\) at \(x = 0\) is

A

\(\frac{1}{1+x}\)

B

\(\frac{1}{1-x}\)

C

\(\frac{1}{(1-x)^2}\)

D

\(\frac{x^3}{3}+1\)

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16

2013 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2013

The solution of the differential equation \(\frac{d^2y}{dx^2} - \frac{dy}{dx} + 0.25y = 0\), given \(y = 0\) at \(x = 0\) and \(\frac{dy}{dx} = 1\) at \(x = 0\) is

A

\(x e^{0.5x} - x e^{-0.5x}\)

B

\(0.5x e^x - 0.5x e^{-x}\)

C

\(x e^{0.5x}\)

D

\(-x e^{0.5x}\)

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17

2014 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2014

The integrating factor for the differential equation
\(\frac{dy}{dx} - \frac{y}{1+x} = (1+x)\) is

A

\(\frac{1}{1+x}\)

B

\((1+x)\)

C

\(x(1+x)\)

D

\(\frac{x}{1+x}\)

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18

2014 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2014

The differential equation \(\frac{d^2 y}{dx^2} + x^2 \frac{dy}{dx} + x^3 y = e^x\) is a

A

non-linear differential equation of first degree

B

linear differential equation of first degree

C

linear differential equation of second degree

D

non-linear differential equation of second degree

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19

2017 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2017

For the initial value problem \[ \frac{dx}{dt} = \sin(t), \quad x(0) = 0 \] the value of \( x \) at \( t = \pi/3 \), is __________.

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20

2018 · Chemical Engineering · Engineering Mathematics · Differential Equations

Chemical Engineering (CH) 2018

Consider the following two equations:
\[\frac{dx}{dt} + x + y = 0\]
\[\frac{dy}{dt} - x = 0\]
The above set of equations is represented by

A

\(\frac{d^2 y}{dt^2} - \frac{dy}{dt} - y = 0\)

B

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

C

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

D

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

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