Differential Equations - Engineering Mathematics - Agricultural Engineering Previous Year Questions

1

2008 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2008

Solution of the ordinary differential equation \(\frac{dy}{dx} = \frac{x^2 + y^2}{y}\) is

A

\(y = \sqrt{(2/3)x^3 + 4x}\)

B

\(y = \sqrt{(2/3)x^3 + 4x + k}\)

C

\(y = \sqrt{(2/3)x^3 - 4x + k}\)

D

\(y = (2/3)x^3 + 4x + k\)

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2

2008 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2008

Solution of the differential equation \(\frac{dy}{dx} - 7y = e^x\) is

A

\(e^x (Ce^{6x} - 6^{-1})\)

B

\(e^x [e^{7x} - (6e^x)^{-1}]\)

C

\(e^x (Ce^{5x} - 6^{-1})\)

D

\(e^x [Ce^{5x} + (6e^x)^{-1}]\)

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3

2010 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2010

A particular solution of \( \frac{d^2 y}{dx^2} + 5 \frac{dy}{dx} - 3y = 6 \) is

A

2.0

B

0.5

C

-0.5

D

-2.0

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4

2010 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2010

The partial differential equation \( \frac{\partial^2 u}{\partial x^2} - 7 \frac{\partial^2 u}{\partial x \partial y} + 2 \frac{\partial^2 u}{\partial y^2} = 0 \) is said to be

A

parabolic

B

hyperbolic

C

elliptic

D

eccentric

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5

2011 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2011

The differential equation \(2\frac{d^2z}{dx^2} + \frac{dz}{dx} + 3y = \sin x\) is considered to be ordinary, as it has

A

one dependent variable

B

more than one dependent variable

C

one independent variable

D

more than one independent variable

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6

2011 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2011

The solution of \(x \frac{dy}{dx} = x^2 + 3y\) for \(x > 0\) and \(y(1) = 2\) is

A

y = x^3 + 3x^2

B

y = x^2 + 3x^3

C

y = -x^2 + 3x^3

D

y = -x^3 - 3x^2

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7

2016 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2016

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

A

\(C = \frac{1}{4} e^{2y} - \frac{1}{3} (e^{3x} + x^3) \)

B

\(C = e^{2y} - \frac{1}{3} (e^{3x} + x^2) \)

C

\(C = \frac{1}{2} e^{2y} - \frac{1}{3} (e^{3x} + x^3) \)

D

\(C = e^{2y} - \frac{1}{3} (e^{3x} + x^3) \)

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8

2018 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2018

The general solution to the second order linear homogeneous differential equation \(y'' - 6y' + 25y = 0\) is

A

\(e^{3x}(a \cos 4x + b \sin 4x)\)

B

\(e^{3ix}(a \cos 4x + b \sin 4x)\)

C

\(e^{4x}(a \cos 3x + b \sin 3x)\)

D

\(e^{4ix}(a \cos 3x + b \sin 3x)\)

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9

2019 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2019

General solution to the differential equation \( y'' + 4y' + 5y = 0 \) is

A

\(e^{2x}(a \cos x + b \sin x) \)

B

\(e^{-2x}(a \cos x + b \sin x) \)

C

\(e^{x}(a \cos 2x + b \sin 2x) \)

D

\(e^{-x}(a \cos 2x + b \sin 2x) \)

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10

2020 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2020

General solution to the ordinary differential equation,
\( \frac{d^3 y}{dx^3} - 6 \frac{d^2 y}{dx^2} + 11 \frac{dy}{dx} - 6y = 0 \) is

A

\(C_1 e^x + C_2 e^{2x} + C_3 e^{3x} \)

B

\(C_1 e^x + C_2 e^{2x} + C_3 e^{3x} \)

C

\(C_1 e^{-x} + C_2 e^{-2x} + C_3 e^{-3x} \)

D

\(C_1 e^{x} + C_2 e^{2x} + C_3 e^{-3x} \)

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11

2021 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2021

Solution of the differential equation \( y'' + y' + 0.25y = 0 \) with the initial values \( y(0) = 3.0 \) and \( y'(0) = -3.5 \) is

A

\(y = (3 - 2x) e^{0.5x} \)

B

\(y = (3 - 2x) e^{-0.25x} \)

C

\(y = (3 - 2x) e^{-0.5x} \)

D

\(y = (2 - 3x) e^{-0.5x} \)

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12

2023 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2023

\( y = a e^{mx} + b e^{-mx} \) is the solution of the differential equation

A

\(\frac{dy}{dx} - my = 0 \)

B

\(\frac{dy}{dx} + my = 0 \)

C

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

D

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

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13

2024 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2024

The complete solution for the differential equation \[ \frac{d^2y}{dx^2} + 2 \frac{dy}{dx} - 3y = 0 \] for \[ y(0) = 0 \] and \[ \frac{dy}{dx}(0) = 8 \] is __________.

A

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

B

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

C

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

D

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

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14

2025 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2025

\(y'' + p(x)y' + q(x)y = r(x)\) is a________ordinary differential equation.

A

second order, nonhomogeneous, and linear

B

second order, homogeneous, and linear

C

second order, homogeneous, and nonlinear

D

first order, nonhomogeneous, and linear

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15

2025 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2025

The value of \( y \) at \( x = 0.3 \) and \( y(x = 0) = 0 \) for the differential equation \( \frac{dy}{dx} = 3y + 2e^{3x} \) is ________. (Rounded off to 3 decimal places)

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16

2026 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2026

Identify the following partial differential equations in the given order based on their generic form where \(c\) is a constant, and \(u_1\) and \(u_2\) are functions of \((x, t)\) and \((x, y)\), respectively:
(i) \(\frac{\partial^2 u_1}{\partial t^2} = c^2 \frac{\partial^2 u_1}{\partial x^2}\)    (ii) \(\frac{\partial u_1}{\partial t} = c^2 \frac{\partial^2 u_1}{\partial x^2}\)    (iii) \(\frac{\partial^2 u_2}{\partial x^2} + \frac{\partial^2 u_2}{\partial y^2} = 0\)

A

(i) Wave equation    (ii) Heat equation    (iii) Laplace equation

B

(i) Heat equation    (ii) Laplace equation    (iii) Wave equation

C

(i) Laplace equation    (ii) Wave equation    (iii) Heat equation

D

(i) Heat equation    (ii) Wave equation    (iii) Laplace equation

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17

2026 · Agricultural Engineering · Engineering Mathematics · Differential Equations

Agricultural Engineering (AG) 2026

The general solution to the ordinary differential equation \(\frac{d^2 y}{dx^2} = \cos 2x\) is

A

\(y = -\frac{\cos 2x}{4} + C_1 x + C_2\), where \(C_1\) and \(C_2\) are constants

B

\(y = \frac{\sin 2x}{2} + C_1\), where \(C_1\) is a constant

C

\(y = \frac{\cos 2x}{4} + C_1 x + C_2\), where \(C_1\) and \(C_2\) are constants

D

\(y = -\frac{\sin 2x}{2} + C_1\), where \(C_1\) is a constant

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