Differential Equations - Engineering Mathematics - Petroleum Engineering Previous Year Questions

1

2016 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2016

For the differential equation \(x^2 \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0\) the general solution is

A

\(y = C_1 x + C_2 e^x\)

B

\(y = C_1 \sin x + C_2 \cos x\)

C

\(y = C_1 e^x + C_2 e^{-x}\)

D

\(y = C_1 x^2 + C_2 x\)

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2

2017 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2017

The roots of the equation \(\frac{d^3 y}{dx^3} - 6 \frac{d^2 y}{dx^2} + 11 \frac{dy}{dx} - 6y = 0\) are:

A

1, 1, 2

B

1, 2, 3

C

1, 3, 4

D

1, 2, 4

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3

2017 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2017

The differential equation \(2xy dx + (1 + x^2) dy = 0\), in which x is an independent variable and y is the dependent variable, is:

A

An ordinary differential equation of second order.

B

A first order nonlinear differential equation.

C

An exact differential equation.

D

A partial differential equation.

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4

2017 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2017

The temperature time profile for a system is given as follows: \(\frac{dT}{dt} + 5T = 500\), where \(T\) is temperature in °C, and \(t\) is time in hours. The initial conditions are \(T(0)=500\)°C. The temperature of the system after 1 hour is ______ °C. (write answer with two decimal places)

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5

2018 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2018

Which one of the following is the integrating factor (IF) for the differential equation, \( (\cos^2 x) \frac{dy}{dx} + y = \cos x \)?

A

\(e^{\tan x} \)

B

\(e^{\cos x} \)

C

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

D

\(e^{\sin x} \)

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6

2018 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2018

The variation of the amount of salt in a tank with time is given by,
\[ \frac{dx}{dt} + 0.025x = 20 \,,\]
where, \( x \) is the amount of salt in kg and \( t \) is the time in minutes. Given that there is no salt in the tank initially, the time at which the amount of salt increases to 200 kg is ______________ minutes. (rounded-off to two decimal places)

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7

2019 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2019

If roots of the auxiliary equation of \( \frac{d^2y}{dx^2} + a\frac{dy}{dx} + by = 0 \) are real and equal, the general solution of the differential equation is

A

y = c₁e^-ax/2 + c₂e^ax/2

B

y = (c₁+c₂x) e^-ax/2

C

y = (c₁+c₂ ln x) e^-ax/2

D

y = (c₁ cos x + c₂ sin x) e^-ax/2

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8

2019 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2019

General solution of the Cauchy-Euler equation \(x^{2}\frac{d^{2}y}{dx^{2}}-7x\frac{dy}{dx}+16y=0\) is

A

\(y=c_{1}x^{2}+c_{2}x^{4}\)

B

\(y=c_{1}x^{2}+c_{2}x^{-4}\)

C

\(y=(c_{1}+c_{2}\ln x)x^{4}\)

D

\(y=c_{1}x^{4}+c_{2}x^{-4}\ln x\)

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9

2020 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2020

The solution of the differential equation, \( \frac{dy}{dx} + \frac{y}{x} = x, (x \neq 0) \) with the condition \( y = 1 \) at \( x = 1 \), is given by

A

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

B

\(y = \frac{x}{2} + \frac{1}{2x} \)

C

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

D

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

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10

2021 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2021

Given the second order ordinary differential equation: $y'' + 3y' - 4y = 0$ with the initial conditions $y(0) = 3$, and $y'(0) = -7$, the value of $y(1)$ is __________ (round off to two decimal places).

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11

2022 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2022

The following second order ordinary differential equation has the boundary conditions: \( y(0) = 0 \), and \( y(1) = 1 \). \[ \frac{d^2 y}{dx^2} + \frac{dy}{dx} = 5y \] The type of above boundary conditions is

A

Neumann

B

Dirichlet

C

Cauchy

D

Robin

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12

2024 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2024

The solution of the initial value problem given by \( y'' + y' - 2y = 0; y(0) = 3, y'(0) = 6 \) is

A

\(4e^x + e^{-2x} \)

B

\(4e^x - e^{-2x} \)

C

\(4e^x + 3e^{-2x} \)

D

\(4e^{-2x} - 3e^x \)

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13

2024 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2024

Which ONE of the following is the implicit form of the solution for the differential equation given below?
\(\frac{dy}{dx} + \frac{(2x+3y)}{(3x+5y)} = 0\).
Note: C in the options below is the integration constant.

A

\(x^2 - 3xy - \frac{5y^2}{2} - C = 0\)

B

\(x^2 - 3xy + \frac{5y^2}{2} - C = 0\)

C

\(x^2 + 3xy - \frac{5y^2}{2} - C = 0\)

D

\(x^2 + 3xy + \frac{5y^2}{2} - C = 0\)

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14

2025 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2025

If \( \frac{dy}{dx} + y = x \), and \( y(0) = 0 \), then the value of \( y(1) \) is

A

\(e \)

B

\(\frac{1}{e} \)

C

1

D

\(-1 \)

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15

2026 · Petroleum Engineering · Engineering Mathematics · Differential Equations

Petroleum Engineering (PE) 2026

For a radioactive material, if N is the number of nuclei present at time t, and λ is decay constant, then which of the following is/are CORRECT representation(s) of the rate of decay?

A

- \(\frac{dN}{dt} = \lambda N^2\)

B

- \(\frac{dN}{dt} = \lambda N\)

C

- \(\frac{dN}{dt} = \frac{\lambda}{N^2}\)

D

- \(\frac{dN}{dt} = \frac{\lambda}{N}\)

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