First-order and Higher-order Equations - Ordinary Differential Equations - Engineering Sciences Previous Year Questions

1

2008 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2008

If the solution of the differential equation \(\frac{dy}{dx} + P(x)y = xy^3\) is \(y^2(1 + ce^{x^2}) = 1\), \(c\) being an arbitrary constant, then \(P(x)\) is

A

\(-x\)

B

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

C

\(x\)

D

\(2x\)

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2

2008 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2008

An LC circuit is shown in the figure. The inductor current, i, when the switch S is opened at t = 0 is best represented by

A

Graph A

B

Graph B

C

Graph C

D

Graph D

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3

2008 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2008

One of the values of \( \frac{1}{(4x^2 D^2 + 8xD + 1)} (\ln x) \) where \( D = \frac{d}{dx} \), is

A

\(\ln x + 4 \)

B

\(\ln x - 4 \)

C

\(4 \ln x - 4 \)

D

\(4 \ln x + 4 \)

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4

2008 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2008

A particular integral of the differential equation \( \frac{d^2 y}{dx^2} - y = \sec h \, x \) is

A

\(-(\cosh x) (\ln \cosh x) + x \sinh x \)

B

\(-(\sinh x) (\ln \cosh x) + x \cosh x \)

C

\((\cosh x) (\ln \sinh x) + x \sinh x \)

D

\((\sinh x) (\ln \cosh x) - x \cosh x \)

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5

2008 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2008

The solution of the first order differential equation \(0 \le x < 1\) \(\frac{dy}{dx} - y^2 = 0\) with \(y(0) = 1\) is

A

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

B

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

C

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

D

\(\frac{x^2}{3} + 1\)

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6

2008 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2008

For the initial value problem \(\frac{dy}{dx} + y = 0\), \(y(0) = 1\), \(y_1\) is the computed value of \(y\) at \(x = 0.2\) obtained by using Euler's method with step size \(h = 0.1\). Then,

A

\(y_1 < e^{-0.2}\)

B

\(e^{-0.2} < y_1 < 1\)

C

\(1 < y_1\)

D

\(y_1 = e^{-0.2}\)

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7

2008 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2008

Consider the initial value problem \(\frac{dy}{dx} = y + x\) with \(y(0) = 2\). The value of \(y(0.1)\) obtained using the fourth order Runge-Kutta method with step size \(h = 0.1\) is

A

2.20000

B

2.21500

C

2.21551

D

2.21576

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8

2009 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2009

Let y1(x) and y2(x) be two linearly independent solutions of \[ \frac{d^2 y}{dx^2} + \frac{6}{x} \frac{dy}{dx} + q(x) y = 0, x \in (1,3) \], where q(x) is a continuous function in (1,3). If the Wronskian of y1(x) and y2(x), at x = 1, denoted by w(y1, y2)(1), is 1, then w(y1, y2)(2) is

A

\(\frac{1}{2^6}\)

B

\(\frac{1}{2^3}\)

C

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

D

2

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9

2010 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2010

Which one of the following is a particular solution of the ordinary differential equation \(x^2 \frac{d^2 y}{dx^2} - \frac{dy}{dx} = 2 x^2 f(x)\) ?

A

\(x^2 \int x f(x) dx + \int x^3 f(x) dx\)

B

\(x^2 \int f(x) dx + \int x^2 f(x) dx\)

C

\(x^2 \int x f(x) dx - \int x^3 f(x) dx\)

D

\(x^2 \int f(x) dx - \int x^2 f(x) dx\)

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10

2011 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2011

A solution of the differential equation $\frac{d^2 y}{dx^2} - 5\frac{dy}{dx} + 6y = 36x$ is

A

$e^{2x} + e^{3x} + 6x + 5$

B

$e^{-3x} + 6x + 5$

C

$e^{2x} + e^{-3x} + 6x + \frac{5}{6}$

D

$e^{2x} + e^{3x} + x + \frac{5}{6}$

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11

2012 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2012

The general solution of \(\frac{d^4 y}{dx^4} - 2\frac{d^3 y}{dx^3} + 2\frac{d^2 y}{dx^2} - 2\frac{dy}{dx} + y = 0\) is

A

\(c_1 e^x + c_2 x e^x + c_3 \cosh(x) + c_4 \sinh(x)\)

B

\(c_1 e^x + c_2 e^{-x} + c_3 e^{ix} + c_4 e^{-ix}\)

C

\(c_1 e^x + c_2 x e^x + c_3 \cos(x) + c_4 \sin(x)\)

D

\(c_1 e^x + c_2 x e^{-x} + c_3 e^{ix} + c_4 e^{-ix}\)

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12

2013 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2013

The general solution of the differential equation \[ x^3 \frac{d^3 y}{d x^3} + x^2 \frac{d^2 y}{d x^2} + x \frac{d y}{d x} - y = 0, \; x > 0 \] is

A

\(C_1 e^x + e^{x/2} \left\{ C_2 \cos \left( \frac{\sqrt{3}}{2} x \right) + C_3 \sin \left( \frac{\sqrt{3}}{2} x \right) \right\} \)

B

\(C_1 x + x^{-1/2} \left\{ C_2 \cos \left( \frac{\sqrt{3}}{2} \log_e x \right) + C_3 \sin \left( \frac{\sqrt{3}}{2} \log_e x \right) \right\} \)

C

\(C_1 e^x + e^{-x/2} \left\{ C_2 \cos \left( \frac{\sqrt{3}}{2} x \right) + C_3 \sin \left( \frac{\sqrt{3}}{2} x \right) \right\} \)

D

\(C_1 x + x^{1/2} \left\{ C_2 \cos \left( \frac{\sqrt{3}}{2} \log_e x \right) + C_3 \sin \left( \frac{\sqrt{3}}{2} \log_e x \right) \right\} \)

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13

2014 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2014

Which of the following is a solution of the differential equation x²y'' + xy' + y = 4 sin(ln x), x > 0?

A

y = 2x sin(ln x)

B

y = -2x sin(ln x)

C

y = -2 ln x cos(ln x)

D

y = 2 ln x cos(ln x)

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14

2014 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2014

An integrating factor of the differential equation $(3x^2 y^3 e^y + y^3 + y^2) dx + (x^3 y^3 e^y - xy) dy = 0$ is

A

$\frac{1}{y}$

B

$\frac{1}{y^2}$

C

$\frac{1}{y^3}$

D

$\ln y$

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15

2015 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2015

The type of the differential equation \((1-x)\frac{d^3y}{dx^3} - \sqrt{1+\left(\frac{dy}{dx}\right)^2} + 5y = \cos(x)\) is

A

linear and first order.

B

non-linear and first order.

C

linear and third order.

D

non-linear and third order.

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16

2015 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2015

The type of the differential equation \( (1-x)\frac{d^3y}{dx^3} - \sqrt{1+(\frac{dy}{dx})^2} +5y = \cos(x) \) is

A

linear and first order.

B

non-linear and first order.

C

linear and third order.

D

non-linear and third order.

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17

2015 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2015

The general solution, $y(x)$, for the differential equation $x \frac{d^2 y}{d x^2} - \frac{dy}{dx} - 1 = 0$ is ($c_1$ and $c_2$ are real constants)

A

$c_1 \frac{x^2}{2} + 2x + c_2$

B

$c_1 \frac{x^2}{2} - x + c_2$

C

$c_1 \frac{x^2}{2} + x + c_2$

D

$c_1 \frac{x^2}{2} - 2x + c_2$

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18

2016 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2016

Let \(y(x)\) be the solution of the initial value problem \(\frac{dy}{dx} + 2xy = x;\ y(0) = 0\). Find the value of \(\lim_{x \to \infty} y(x)\).

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19

2017 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2017

Consider the ordinary differential equation \[ y'' + \alpha y' + \beta y = 0, \] where \( \alpha \) and \( \beta \) are constants. If \( y(x) = x e^x \) is a solution of the above equation, then the value of \( \beta - \alpha \) is ______________.

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20

2017 · Engineering Sciences · Ordinary Differential Equations · First-order and Higher-order Equations

Engineering Sciences (XE) 2017

Consider the ordinary differential equation \(x^2 y'' + xy' - y = x, \quad x > 0\). In terms of arbitrary constants \(c_1\) and \(c_2\), the general solution of the above equation is

A

\(y(x) = c_1 x + c_2 x^{-1} + x^3\)

B

\(y(x) = c_1 x^2 + c_2 x^{-1} + \frac{1}{2} x\)

C

\(y(x) = c_1 x + c_2 x^{-1} + \frac{1}{2} x \ln x\)

D

\(y(x) = c_1 x + c_2 + x^{-1}\)

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