Differential Equations - Engineering Mathematics - Instrumentation Engineering Previous Year Questions

1

2010 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2010

Consider the differential equation \(\frac{dy}{dx} + y = e^x\) with \(y(0) = 1\). The value of \(y(1)\) is

A

\(e + e^{-1}\)

B

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

C

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

D

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

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2

2011 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2011

Consider the differential equation \(\ddot{y} + 2\dot{y} + y = 0\) with boundary conditions \(y(0) = 1, y(1) = 0\). The value of \(y(2)\) is

A

-1

B

\(-e^{-1}\)

C

\(-e^{-2}\)

D

\(-e^{-2}\)

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3

2013 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2013 [Session 1]

The type of the partial differential equation \(\frac{\partial f}{\partial t} = \frac{\partial^2 f}{\partial x^2}\) is

A

Parabolic

B

Elliptic

C

Hyperbolic

D

Nonlinear

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4

2013 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2013 [Session 1]

The maximum value of the solution \(y(t)\) of the differential equation \(y(t) + \ddot{y}(t) = 0\) with initial conditions \(\dot{y}(0) = 1\) and \(y(0) = 1\), for \(t \geq 0\) is

A

1

B

2

C

\(\pi\)

D

\(\sqrt{2}\)

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5

2013 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2013 [Session 2]

The maximum value of the solution \( y(t) \) of the differential equation \( y(t)+\dot{y}(t)=0 \) with initial conditions \( \dot{y}(0)=1 \) and \( y(0)=1 \), for \( t \geq 0 \) is

A

1

B

2

C

\(\pi\)

D

\(\sqrt{2}\)

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6

2013 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2013 [Session 3]

The type of the partial differential equation \frac{\partial f}{\partial t} = \frac{\partial^2 f}{\partial x^2} is

A

Parabolic

B

Elliptic

C

Hyperbolic

D

Nonlinear

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7

2013 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2013 [Session 3]

The maximum value of the solution \(y(t)\) of the differential equation \(y(t)+\dot{y}(t)=0\) with initial conditions \(\dot{y}(0)=1\) and \(y(0)=1\), for \(t \geq 0\) is

A

1

B

2

C

π

D

√2

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8

2013 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2013 [Session 4]

The maximum value of the solution \(y(t)\) of the differential equation \(y(t)+\dot{y}(t)=0\) with initial conditions \(\dot{y}(0)=1\) and \(y(0)=1\), for \(t \ge 0\) is

A

1

B

2

C

π

D

\(\sqrt{2}\)

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9

2014 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2014

The figure shows the plot of \(y\) as a function of \(x\)

The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is:

A

\(\frac{d^2 y}{dx^2} = 1\)

B

\(\frac{dy}{dx} = x\)

C

\(\frac{dy}{dx} = -x\)

D

\(\frac{dy}{dx} = |x|\)

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10

2024 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2024

The solution of an ordinary differential equation \( y''' + 3y'' + 3y' + y = 30e^{-t} \) is \( y(t) = (c_0 + c_1 t - c_2 t^2 + c_3 t^3)e^{-t} \). Given \( y(0) = 3, y'(0) = -3 \) and \( y''(0) = -47 \), the value of \( (c_0 + c_1 + c_2 + c_3) \) is _____ (rounded off to nearest integer).
Note: \( y''' = d^3y/dt^3, y'' = d^2y/dt^2, y' = dy/dt \) and \( c_0, c_1, c_2, c_3 \) are constants.

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11

2025 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2025

The solution of the differential equation \(\frac{dy}{dx} = 9\frac{x}{y}\) represents

A

a hyperbola

B

a parabola

C

an ellipse

D

a circle

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12

2007 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2007

The boundary-value problem \(y'' + \lambda y = 0\), \(y(0) = y(\pi) = 0\) will have non-zero solutions if and only if the values of \(\lambda\) are

A

\(0, \pm 1, \pm 2, \cdots\)

B

\(1, 2, 3, \cdots\)

C

\(1, 4, 9, \cdots\)

D

\(1, 9, 25, \cdots\)

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13

2008 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2008

Consider the differential equation \( \frac{dy}{dx} = 1 + y^2 \). Which one of the following can be a particular solution of this differential equation?

A

\(y = \tan(x + 3) \)

B

\(y = \tan x + 3 \)

C

\(x = \tan(y + 3) \)

D

\(x = \tan y + 3 \)

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14

2020 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2020

Consider the differential equation \(\frac{dx}{dt} = \sin(x)\), with the initial condition \(x(0) = 0\). The solution to this ordinary differential equation is ____

A

\(x(t) = 0\)

B

\(x(t) = \sin(t)\)

C

\(x(t) = \cos(t)\)

D

\(x(t) = \sin(t) - \cos(t)\)

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15

2022 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2022

Consider the differential equation \[\frac{dy}{dx} + y \ln(y) = 0\] If y(0) = e, then y(1) is ________

A

e^e

B

e^{-e}

C

e^{(1/e)}

D

e^{(-1/e)}

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16

2023 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2023

The solution \(x(t), t \geq 0\), to the differential equation \[\ddot{x} = -kx, k > 0\] with initial conditions \(x(0) = 1\) and \(\dot{x}(0) = 0\) is

A

\(x(t) = 2e^{-kt} + 2kt - 1\)

B

\(x(t) = 2e^{-kt} - 1\)

C

\(x(t) = 1\)

D

\(x(t) = 2e^{-kt} - kt - 1\)

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17

2026 · Instrumentation Engineering · Engineering Mathematics · Differential Equations

Instrumentation Engineering (IN) 2026

Given a differential equation \(\frac{d^2x}{dt^2} + x = 0\) with \(x(0) \neq 0\).
Which of the following statements is/are true?

A

\(|x(t)|^2 + |\frac{dx(t)}{dt}|^2 = 0\) for all \(t \geq 0\)

B

\(|x(t)|^2 + |\frac{dx(t)}{dt}|^2 = c\) for all \(t \geq 0\), for some real constant \(c > 0\)

C

\(|x(t)|^2 + |\frac{dx(t)}{dt}|^2 = e^{jt}\) for all \(t \geq 0\)

D

\(|x(t)|^2 + |\frac{dx(t)}{dt}|^2 = \sin t\) for all \(t \geq 0\)

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