Ordinary Differential Equation (ODE) - Engineering Mathematics - Civil Engineering Previous Year Questions

1

2014 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2014 [Session 2]

The integrating factor for the differential equation \( \frac{dP}{dt} + k_2 P = k_1 L_0 e^{-k_1 t} \) is

A

\(e^{-k_1 t} \)

B

\(e^{-k_2 t} \)

C

\(e^{k_1 t} \)

D

\(e^{k_2 t} \)

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2

2016 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2016 [Session 1]

The respective expressions for complimentary function and particular integral part of the solution of the differential equation \(\frac{d^4 y}{dx^4} + 3\frac{d^2 y}{dx^2} = 100x^2\) are

A

\([c_1 + c_2 x + c_3 \sin \sqrt{3}x + c_4 \cos \sqrt{3}x]\) and \([3x^2 - 12x^2 + c]\)

B

\([c_2 x + c_3 \sin \sqrt{3}x + c_4 \cos \sqrt{3}x]\) and \([5x^4 - 12x^2 + c]\)

C

\([c_1 + c_2 \sin \sqrt{3}x + c_4 \cos \sqrt{3}x]\) and \([3x^4 - 12x^2 + c]\)

D

\([c_1 + c_2 x + c_3 \sin \sqrt{3}x + c_4 \cos \sqrt{3}x]\) and \([5x^4 - 12x^2 + c]\)

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3

2017 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2017 [Session 2]

Consider the following second-order differential equation: \( y'' - 4y' + 3y = 2t - 3t^2 \). The particular solution of the differential equation is

A

\(-2 - 2t - t^2 \)

B

\(-2t - t^2 \)

C

\(2t - 3t^2 \)

D

\(-2 - 2t - 3t^2 \)

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4

2018 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2018 [Session 2]

The solution of the equation \(x \frac{dy}{dx} + y = 0\) passing through the point (1,1) is

A

\(x\)

B

\(x^2\)

C

\(x^{-1}\)

D

\(x^{-2}\)

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5

2019 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2019 [Session 2]

An ordinary differential equation is given below: \[\left(\frac{dy}{dx}\right)(x\ln x) = y\] The solution for the above equation is (Note: K denotes a constant in the options)

A

y = Kx\ln x

B

y = Kxe^x

C

y = Kxe^{-x}

D

y = K\ln x

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6

2020 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2020 [Session 2]

The ordinary differential equation \( \frac{d^2u}{dx^2} - 2x^2u + \sin x = 0 \) is

A

linear and homogeneous

B

linear and nonhomogeneous

C

nonlinear and homogeneous

D

nonlinear and nonhomogeneous

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7

2020 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2020 [Session 2]

An ordinary differential equation is given below:
\( 6\frac{d^2 y}{dx^2} + \frac{dy}{dx} - y = 0 \)
The general solution of the above equation (with constants \( C_1 \) and \( C_2 \)), is

A

\(y(x) = C_1 e^{-\frac{x}{3}} + C_2 e^{\frac{x}{2}} \)

B

\(y(x) = C_1 e^{\frac{x}{3}} + C_2 e^{-\frac{x}{2}} \)

C

\(y(x) = C_1 x e^{-\frac{x}{3}} + C_2 e^{\frac{x}{2}} \)

D

\(y(x) = C_1 e^{-\frac{x}{3}} + C_2 x e^{\frac{x}{2}} \)

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8

2024 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2024 [Session 1]

A 2 m × 2 m tank of 3 m height has inflow, outflow and stirring mechanisms. Initially, the tank was half-filled with fresh water. At \(t = 0\), an inflow of a salt solution of concentration 5 g/m³ at the rate of 2 litre/s and an outflow of the well stirred mixture at the rate of 1 litre/s are initiated. This process can be modelled using the following differential equation: \[ \frac{dm}{dt} + \frac{m}{6000 + t} = 0.01 \] where \(m\) is the mass (grams) of the salt at time \(t\) (seconds). The mass of the salt (in grams) in the tank at 75% of its capacity is __________ (rounded off to 2 decimal places).

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9

2024 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2024 [Session 2]

Consider two Ordinary Differential Equations (ODEs):
P: \( \frac{dy}{dx} = \frac{x^4 + 3x^2y^2 + 2y^4}{x^3y} \)
Q: \( \frac{dy}{dx} = \frac{-y^2}{x^2} \)
Which one of the following options is CORRECT?

A

P is a homogeneous ODE and Q is an exact ODE.

B

P is a homogeneous ODE and Q is not an exact ODE.

C

P is a nonhomogeneous ODE and Q is an exact ODE.

D

P is a nonhomogeneous ODE and Q is not an exact ODE.

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10

2025 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2025 [Session 1]

Let y be the solution of the initial value problem y'' + 0.8y' + 0.16y = 0, where y(0) = 3 and y'(0) = 4.5. Then, y(1) is equal to ______ (rounded off to 1 decimal place).

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11

2025 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2025 [Session 2]

The “order” of the following ordinary differential equation is ______. \[ \frac{d^3y}{dx^3} + \left(\frac{d^2y}{dx^2}\right)^6 + \left(\frac{dy}{dx}\right)^4 + y = 0 \]

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12

2025 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2025 [Session 2]

Pick the CORRECT solution for the following differential equation
\[ \frac{dy}{dx} = e^{x-y} \]

A

y = ln(e^x + Constant)

B

ln(y) = x + Constant

C

ln(y) = ln(e^x) + Constant

D

y = x + Constant

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13

2008 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2008

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

A

\(y = P \cos x + Q \sin x \)

B

\(y = P \cos x \)

C

\(y = P \sin x \)

D

\(y = P \sin^2 x \)

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14

2008 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2008

Solution of \( \frac{dy}{dx} = - \frac{x}{y} \) at x = 1 and y = \( \sqrt{3} \) is

A

\(x - y^2 = -2 \)

B

\(x + y^2 = 4 \)

C

\(x^2 - y^2 = -2 \)

D

\(x^2 + y^2 = 4 \)

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15

2009 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2009

Solution of the differential equation 3y\frac{dy}{dx} + 2x = 0 represents a family of

A

ellipses

B

circles

C

parabolas

D

hyperbolas

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16

2010 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2010

The order and degree of the differential equation \[\frac{d^3 y}{dx^3} + 4 \left( \frac{dy}{dx} \right)^3 + y^2 = 0\] are respectively

A

3 and 2

B

2 and 3

C

3 and 3

D

3 and 1

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17

2010 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2010

The solution to the ordinary differential equation \[ \frac{d^2 y}{dx^2} + \frac{dy}{dx} - 6y = 0 \] is

A

\(y = c_1 e^{3x} + c_2 e^{-2x} \)

B

\(y = c_1 e^{3x} + c_2 e^{2x} \)

C

\(y = c_1 e^{-3x} + c_2 e^{2x} \)

D

\(y = c_1 e^{-3x} + c_2 e^{-2x} \)

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18

2011 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2011

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

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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19

2012 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2012

The solution of the ordinary differential equation \( \frac{d y}{d x} + 2 y = 0 \) for the boundary condition, \( y = 5 \) at \( x = 1 \) is

A

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

B

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

C

\(y = 10.95 e^{-2x} \)

D

\(y = 36.95 e^{-2x} \)

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20

2026 · Civil Engineering · Engineering Mathematics · Ordinary Differential Equation (ODE)

Civil Engineering (CE) 2026

An ordinary differential equation is given below.
\[ x^2 \frac{d^2 y}{dx^2} = 6y \]
Considering \( a \) and \( b \) as arbitrary constants, the general solution of the equation is

A

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

B

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

C

\(y(x) = ax^2 + b\ln x \)

D

\(y(x) = ax^3 + b\ln x \)

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