Differential Equations - Engineering Mathematics - Metallurgical Engineering Previous Year Questions

1

2008 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2008

Which of the following is a solution for \(\frac{\partial z}{\partial t} = \frac{\partial^2 z}{\partial x^2}\)

A

\(z(x,t) = [A \sin x]e^{-j\lambda t}\)

B

\(z(x,t) = [A \sin(\lambda x)] e^{-\lambda^2 t}\)

C

\(z(x,t) = \frac{A}{t} e^{-x^2/t}\)

D

\(z(x,t) = [B \cos(\lambda x)] \sqrt{t}\)

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2

2009 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2009

The solution function \(y = f(x)\) for the ordinary differential equation, \(dy/dx = 3x^2 - 2x\), passes through (1,1). The magnitude of \(y\) at \(x = 3\) is

A

0

B

18

C

19

D

21

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3

2010 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2010

Which of the following is typical form of a wave equation?

A

\(x^2 \frac{d^2 u}{d x^2} + x \frac{d u}{d x} + u = 0 \)

B

\(\nabla^2 u = \frac{\partial^2 u}{\partial t^2} ; a > 0 \)

C

\(\nabla^2 u = 0 \)

D

\(\nabla^2 u = a \frac{\partial u}{\partial t} ; a > 0 \)

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4

2010 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2010

Solution of the equation \(2x\frac{dy}{dx} + 3y = 0\) is

A

\(x^{-3/2}\)

B

\(x^{-3/2}\)

C

\(x^{-3/2}\)

D

\(x^{3/2}\)

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5

2012 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2012

What does the solution of the following ordinary differential equation represent?
\[ y\frac{dy}{dx} + x = 0 \]

A

A parabola

B

A circle

C

An ellipse

D

A hyperbola

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6

2012 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2012

Amplitude ratio is

A

3

B

1/3

C

-1/3

D

-3

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7

2013 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2013

Degree and order of the differential equation \(\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} - 6y = 0\), respectively, are

A

1 and 2

B

2 and 1

C

1 and 1

D

2 and 2

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8

2014 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2014

Identify the wave equation among the following equations.

A

\(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0\)

B

\(\frac{\partial^2 f}{\partial x^2} = A \frac{\partial f}{\partial t}\)

C

\(\frac{\partial^2 f}{\partial x^2} = A^2 \frac{\partial^2 f}{\partial t^2}\)

D

\(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial x \partial y} + \frac{\partial^2 f}{\partial y^2} = 0\)

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9

2016 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2016

The solution of the differential equation \[ \frac{d^2 y}{dx^2} = \frac{dy}{dx} \] is

A

y = e^x + C

B

y = e^{-x} + C

C

y = C_1 e^{-x} + C_2

D

y = C_1 e^{x} + C_2

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10

2017 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2017

For the second order linear ordinary differential equation, \(\frac{d^2y}{dx^2} + p\frac{dy}{dx} + qy = 0\), the following function is a solution: \(y = e^{\lambda x}\) Which one of the following statements is NOT TRUE?

A

\(\lambda\) has two values: one complex and one real

B

\(\lambda^2 + p\lambda + q = 0\)

C

\(\lambda\) has two real values

D

\(\lambda\) has two complex values

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11

2018 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2018

Consider the following Ordinary Differential Equation: \[ \frac{d}{dx} \left[ c \frac{dc}{dx} \right] = 0 \] In a domain \(0 \le x \le t\), with boundary conditions \(c(0) = 0.5\) and \(c(t) = 1.0\), pick the appropriate choice for \(c(x)\) from the following options:

A

P

B

Q

C

R

D

S

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12

2020 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2020

The general solution to the following homogeneous ODE, \[\frac{d^2 y}{dt^2} + 4\frac{dy}{dt} + 3y = 0,\] is, \[y(t) = C_1 e^{\lambda_1 t} + C_2 e^{\lambda_2 t}.\] The values of \(\lambda_1\) and \(\lambda_2\) are:

A

-1 and -3

B

-3 and -3

C

1 and -3

D

1 and 3

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13

2022 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2022

The general solution to the following differential equation is ______, where \(A\) and \(B\) are constants
\[\frac{d^2y}{dt^2} - 4\frac{dy}{dt} + 4y = 0\]

A

\(y = A \sin(2t) + B\)

B

\(y = A \sin(2t) + B \cos(2t)\)

C

\(y = A e^{2t} + B e^{-2t}\)

D

\(y = A e^{2t} + B t e^{2t}\)

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14

2023 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2023

Order (O) and degree (D) of the differential equation \(\left(\frac{dy}{dx}\right)^3 = \sqrt{\frac{d^2y}{dx^2}} + 10\) are

A

O = 2 and D = 1

B

O = 1 and D = 2

C

O = 6 and D = 1

D

O = 2 and D = 6

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15

2024 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2024

If $\frac{dy}{dx} = 4xy, \quad y(0) = 1$, then

A

$y = 2x^2 + 1$

B

$y = 2e^{2x^2} - 1$

C

$y = 2e^{x^2} - 1$

D

$y = e^{2x^2}$

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16

2025 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2025

For two continuous functions $M(x,y)$ and $N(x,y)$, the relation $M\,dx + N\,dy = 0$ describes an exact differential equation if

A

\(\frac{\partial M}{\partial x} = \frac{\partial N}{\partial y}\)

B

\(\frac{\partial M}{\partial x} = -\frac{\partial N}{\partial y}\)

C

\(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\)

D

\(\frac{\partial M}{\partial y} = -\frac{\partial N}{\partial x}\)

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17

2026 · Metallurgical Engineering · Engineering Mathematics · Differential Equations

Metallurgical Engineering (MT) 2026

Solution of the differential equation \( 10x^2 \frac{d^2 y}{dx^2} - 20x \frac{dy}{dx} + 22.4y = 0 \) is \( y = c_1 x^{m_1} + c_2 x^{m_2} \) where \( m_1 \neq m_2 \). The value of \( m_1 + m_2 \) (answer in integer) is ______________

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