Differential Equations - Engineering Mathematics - Electronics & Communication Engineering Previous Year Questions

1

2012 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2012

With initial condition \(x(1) = 0.5\), the solution of the differential equation, \(t rac{dx}{dt} + x = t\) is

A

\(x = t - rac{1}{2}\)

B

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

C

\(x = rac{t^2}{2}\)

D

\(x = rac{t}{2}\)

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2

2014 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2014 [Session 4]

With initial values y(0) = y'(0) = 1, the solution of the differential equation \[ \frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 4y = 0 \] at x = 1 is _____.

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3

2014 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2014 [Session 4]

If a and b are constants, the most general solution of the differential equation \(\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + x = 0\) is

A

\(ae^{-t}\)

B

\(ae^{-t} + bte^{-t}\)

C

\(ae^t + bte^{-t}\)

D

\(ae^{-2t}\)

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4

2016 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2016 [Session 2]

The ordinary differential equation \( \frac{dx}{dt} = -3x + 2 \), with \( x(0) = 1 \) is to be solved using the forward Euler method. The largest time step that can be used to solve the equation without making the numerical solution unstable is ______

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5

2016 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2016 [Session 3]

The particular solution of the initial value problem given below is
\(\frac{d^2 y}{d x^2}+12 \frac{d y}{d x}+36 y=0\) with \(y(0)=3\) and \(\left.\frac{d y}{d x}\right|_{x=0}=-36\)

A

\((3-18 x) e^{-6 x}\)

B

\((3+25 x) e^{-6 x}\)

C

\((3+20 x) e^{-6 x}\)

D

\((3-12 x) e^{-6 x}\)

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6

2016 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2016 [Session 3]

Consider the first order initial value problem \[ y' = y + 2x - x^2, \quad y(0) = 1, \quad (0 \leq x < \infty) \] with exact solution \( y(x) = x^2 + e^x \). For x = 0.1, the percentage difference between the exact solution and the solution obtained using a single iteration of the second-order Runge-Kutta method with step-size h = 0.1 is __________

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7

2017 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2017

The general solution of the differential equation a2d2y/dx2 + 2dy/dx - 5y = 0 in terms of arbitrary constants K1 and K2 is

A

K₁e^(-1+√6)x + K₂e^(-1-√6)x

B

K₁e^(-1+√8)x + K₂e^(-1-√8)x

C

K₁e^(-2+√6)x + K₂e^(-2-√6)x

D

K₁e^(-2+√8)x + K₂e^(-2-√8)x

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8

2018 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2018

A curve passes through the point \((x = 1, y = 0)\) and satisfies the differential equation \(\frac{dy}{dx} = \frac{x^2 + y^2}{2y} + \frac{y}{x}\). The equation that describes the curve is

A

\(\ln\left(1 + \frac{y^2}{x^2}\right) = x - 1\)

B

\(\frac{1}{2}\ln\left(1 + \frac{y^2}{x^2}\right) = x - 1\)

C

\(\ln\left(1 + \frac{y}{x}\right) = x - 1\)

D

\(\frac{1}{2}\ln\left(1 + \frac{y}{x}\right) = x - 1\)

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9

2018 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2018

The position of a particle y(t) is described by the differential equation:
\[ \frac{d^2 y}{dt^2} = -\frac{dy}{dt} - \frac{5y}{4}. \]
The initial conditions are y(0) = 1 and \[ \frac{dy}{dt}\bigg|_{t=0} = 0 \]. The position (accurate to two decimal places) of the particle at t = π is ______.

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10

2019 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2019

The families of curves represented by the solution of the equation \[\frac{dy}{dx} = -\left(\frac{x}{y}\right)^n\] for \(n = -1\) and \(n = +1\), respectively, are

A

Parabolas and Circles

B

Circles and Hyperbolas

C

Hyperbolas and Circles

D

Hyperbolas and Parabolas

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11

2019 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2019

Consider the homogeneous ordinary differential equation \( x^2 \frac{d^2 y}{dx^2} - 3x \frac{dy}{dx} + 3y = 0, \quad x > 0 \) with y(x) as a general solution. Given that y(1) = 1 and y(2) = 14 the value of y(1.5), rounded off to two decimal places, is ___________.

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12

2020 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2020

Which one of the following options contains two solutions of the differential equation \( \frac{dy}{dx} = (y-1)x \)?

A

\(\ln|y-1| = 0.5x^2 + C \) and \(y = 1 \)

B

\(\ln|y-1| = 2x^2 + C \) and \(y = 1 \)

C

\(\ln|y-1| = 0.5x^2 + C \) and \(y = -1 \)

D

\(\ln|y-1| = 2x^2 + C \) and \(y = -1 \)

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13

2020 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2020

Which one of the following options contains two solutions of the differential equation \[ \frac{dy}{dx} = (y-1)x ? \]

A

\(\ln|y-1| = 0.5x^2 + C\) and \(y = 1\)

B

\(\ln|y-1| = 2x^2 + C\) and \(y = 1\)

C

\(\ln|y-1| = 0.5x^2 + C\) and \(y = -1\)

D

\(\ln|y-1| = 2x^2 + C\) and \(y = -1\)

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14

2020 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2020

The general solution of \(\frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = 0\) is

A

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

B

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

C

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

D

\(y = C_1 e^{3x}\)

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15

2021 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2021

Consider the differential equation given below.
\[ \frac{dy}{dx} + \frac{x}{1-x^2} y = x\sqrt{y} \]
The integrating factor of the differential equation is

A

\((1-x^2)^{-3/4} \)

B

\((1-x^2)^{-1/4} \)

C

\((1-x^2)^{-3/2} \)

D

\((1-x^2)^{-1/2} \)

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16

2022 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2022

Consider the following partial differential equation (PDE)
\[ a\frac{\partial^2 f(x,y)}{\partial x^2} + b\frac{\partial^2 f(x,y)}{\partial y^2} = f(x,y), \]
where \( a \) and \( b \) are distinct positive real numbers. Select the combination(s) of values of the real parameters \( \xi \) and \( \eta \) such that \( f(x,y) = e^{(\xi x + \eta y)} \) is a solution of the given PDE.

A

\(\xi = \frac{1}{\sqrt{2a}} \), \(\eta = \frac{1}{\sqrt{2b}} \)

B

\(\xi = \frac{1}{\sqrt{a}} \), \(\eta = 0 \)

C

\(\xi = 0 \), \(\eta = 0 \)

D

\(\xi = \frac{1}{\sqrt{a}} \), \(\eta = \frac{1}{\sqrt{b}} \)

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17

2022 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2022

Consider the following wave equation, \[\frac{\partial^2 f(x,t)}{\partial t^2} = 10000 \frac{\partial^2 f(x,t)}{\partial x^2}\] Which of the given options is/are solution(s) to the given wave equation?

A

\(f(x,t) = e^{-(x-100t)^2} + e^{-(x+100t)^2}\)

B

\(f(x,t) = e^{-(x-100t)^2} + 0.5e^{-(x+100t)^2}\)

C

\(f(x,t) = e^{-(x-100t)^2} + \sin(x + 100t)\)

D

\(f(x,t) = e^{j100\pi(-100x+t)} + e^{j100\pi(100x+t)}\)

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18

2024 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2024

The general form of the complementary function of a differential equation is given by \( y(t) = (At + B)e^{-2t} \), where \( A \) and \( B \) are real constants determined by the initial condition. The corresponding differential equation is _____.

A

\(\frac{d^2y}{dt^2} + 4\frac{dy}{dt} + 4y = f(t) \)

B

\(\frac{d^2y}{dt^2} + 4y = f(t) \)

C

\(\frac{d^2y}{dt^2} + 3\frac{dy}{dt} + 2y = f(t) \)

D

\(\frac{d^2y}{dt^2} + 5\frac{dy}{dt} + 6y = f(t) \)

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19

2025 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics & Communication Engineering (EC) 2025

The function \( y(t) \) satisfies
\( t^2 y''(t) - 2t y'(t) + 2y(t) = 0, \)
where \( y'(t) \) and \( y''(t) \) denote the first and second derivatives of \( y(t) \), respectively.
Given \( y'(0) = 1 \) and \( y'(1) = -1 \), the maximum value of \( y(t) \) over \( [0,1] \) is _____ (rounded off to two decimal places).

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20

2026 · Electronics & Communication Engineering · Engineering Mathematics · Differential Equations

Electronics and Communication Engineering (EC) 2026

Consider the differential equation \(\dot{W} = A\vec{W}\), with \(\vec{W}(t = 0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}\).
If \(\vec{W}(t) = e^t \vec{u}_x + e^{-2t} \vec{u}_y\) be the solution to the equation where \(\vec{u}_x\) and \(\vec{u}_y\) are unit vectors along the positive x and y axes respectively, then which of the following options is the correct matrix representing \(A\)?

A

\(\begin{bmatrix} 1 & 0 \\ 0 & -2 \end{bmatrix}\)

B

\(\begin{bmatrix} -1 & 0 \\ 0 & 2 \end{bmatrix}\)

C

\(\begin{bmatrix} 0 & -2 \\ 1 & 0 \end{bmatrix}\)

D

\(\begin{bmatrix} 0 & 2 \\ -1 & 0 \end{bmatrix}\)

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