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Recent questions and answers in Differential Equations
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GATE Mechanical 2021 Set 2 | Question: 2
If the Laplace transform of a function $f(t)$ is given by $\frac{s+3}{\left ( s+1 \right )\left ( s+2 \right )}$, then $f(0)$ is $0$ $\frac{1}{2}$ $1$ $\frac{3}{2}$
Hashtag
answered
in
Differential Equations
Jun 20, 2021
by
Hashtag
140
points
gateme-2021-set2
differential-equations
laplace-transforms
0
answers
0
votes
GATE Mechanical 2021 Set 2 | Question: 28
Consider the following differential equation $\left ( 1+y \right )\frac{dy}{dx}=y.$ The solution of the equation that satisfies condition $y(1)=1$ is $2ye^{y}=e^{x}+e$ $y^{2}e^{y}=e^{x}$ $ye^{y}=e^{x}$ $\left ( 1+y \right )e^{y}=2e^{x}$
go_editor
asked
in
Differential Equations
Mar 1, 2021
by
go_editor
5.0k
points
gateme-2021-set2
differential-equations
0
answers
0
votes
GATE Mechanical 2021 Set 1 | Question: 1
If $y(x)$ satisfies the differential equation $(\sin x) \dfrac{\mathrm{d}y }{\mathrm{d} x} + y \cos x = 1,$ subject to the condition $y(\pi/2) = \pi/2,$ then $y(\pi/6)$ is $0$ $\frac{\pi}{6}$ $\frac{\pi}{3}$ $\frac{\pi}{2}$
gatecse
asked
in
Differential Equations
Feb 22, 2021
by
gatecse
1.6k
points
gateme-2021-set1
differential-equations
0
answers
0
votes
GATE Mechanical 2021 Set 1 | Question: 3
The Dirac-delta function $\left ( \delta \left ( t-t_{0} \right ) \right )$ for $\text{t}$, $t_{0} \in \mathbb{R}$ ... $0$ $\infty$ $e^{sa}$ $e^{-sa}$
gatecse
asked
in
Differential Equations
Feb 22, 2021
by
gatecse
1.6k
points
gateme-2021-set1
differential-equations
laplace-transforms
0
answers
0
votes
GATE Mechanical 2021 Set 1 | Question: 4
The ordinary differential equation $\dfrac{dy}{dt}=-\pi y$ subject to an initial condition $y\left ( 0 \right )=1$ ... ___________________. $0< h< \frac{2}{\pi }$ $0< h< 1$ $0< h< \frac{\pi }{2}$ for all $h> 0$
gatecse
asked
in
Differential Equations
Feb 22, 2021
by
gatecse
1.6k
points
gateme-2021-set1
differential-equations
0
answers
0
votes
GATE2020-ME-2: 4
The solution of $\dfrac{d^2y}{dt^2}-y=1,$ which additionally satisfies $y \bigg \vert_{t=0} = \dfrac{dy}{dt} \bigg \vert_{t=0}=0$ in the Laplace $s$-domain is $\dfrac{1}{s(s+1)(s-1)} \\$ $\dfrac{1}{s(s+1)} \\$ $\dfrac{1}{s(s-1)} \\$ $\dfrac{1}{s-1} \\$
go_editor
asked
in
Differential Equations
Sep 18, 2020
by
go_editor
5.0k
points
gateme-2020-set2
differential-equations
laplace-transforms
0
answers
0
votes
GATE2020-ME-1: 3
The Laplace transform of a function $f(t)$ is $L( f )=\dfrac{1}{(s^{2}+\omega ^{2})}.$ Then, $f(t)$ is $f\left ( t \right )=\dfrac{1}{\omega ^{2}}\left ( 1-\cos\:\omega t \right ) \\$ $f\left ( t \right )=\dfrac{1}{\omega}\cos\:\omega t \\$ $f\left ( t \right )=\dfrac{1}{\omega}\sin\:\omega t \\$ $f\left ( t \right )=\dfrac{1}{\omega^{2}}\left ( 1-\sin\:\omega t \right )$
go_editor
asked
in
Differential Equations
Feb 19, 2020
by
go_editor
5.0k
points
gateme-2020-set1
differential-equations
laplace-transforms
1
answer
0
votes
GATE2019 ME-1: 3
For the equation $\dfrac{dy}{dx}+7x^2y=0$, if $y(0)=3/7$, then the value of $y(1)$ is $\dfrac{7}{3}e^{-7/3} \\$ $\dfrac{7}{3}e^{-3/7} \\$ $\dfrac{3}{7}e^{-7/3} \\$ $\dfrac{3}{7}e^{-3/7}$
ankitgupta.1729
answered
in
Differential Equations
May 24, 2019
by
ankitgupta.1729
410
points
gateme-2019-set1
differential-equations
1
answer
0
votes
GATE2019 ME-2: 3
The differential equation $\dfrac{dy}{dx}+4y=5$ is valid in the domain $0 \leq x \leq 1$ with $y(0)=2.25$. The solution of the differential equation is $y=e^{-4x}+5$ $y=e^{-4x}+1.25$ $y=e^{4x}+5$ $y=e^{4x}+1.25$
ankitgupta.1729
answered
in
Differential Equations
May 24, 2019
by
ankitgupta.1729
410
points
gateme-2019-set2
differential-equations
0
answers
0
votes
GATE2019 ME-1: 27
A harmonic function is analytic if it satisfies the Laplace equation. If $u(x,y)=2x^2-2y^2+4xy$ is a harmonic function, then its conjugate harmonic function $v(x,y)$ is $4xy-2x^2+2y^2+ \text{constant}$ $4y^2-4xy + \text{constant}$ $2x^2-2y^2+ xy + \text{constant}$ $-4xy+2y^2-2x^2+ \text{constant}$
Arjun
asked
in
Differential Equations
Feb 9, 2019
by
Arjun
27.4k
points
gateme-2019-set1
differential-equations
laplace-transforms
0
answers
0
votes
GATE ME 2012 | Question: 46
Consider the differential equation $x^2 \dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}-4y=0$ with the boundary conditions of $y(0)=0$ and $y(1)=1$. The complete solution of the differential equation is $x^2 \\$ $\sin \left (\dfrac{\pi x}{2} \right ) \\$ $e^x \sin \left (\dfrac{\pi x}{2} \right ) \\$ $e^{-x} \sin \left (\dfrac{\pi x}{2} \right) \\$
Andrijana3306
asked
in
Differential Equations
Mar 20, 2018
by
Andrijana3306
1.5k
points
gateme-2012
differential-equation
boundary-value-problems
0
answers
0
votes
GATE2018-2-36
Given the ordinary differential equation $\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-6y=0$ with $y(0)=0$ and $\dfrac{dy}{dx}(0)=1$, the value of $y(1)$ is __________ (correct to two decimal places).
Arjun
asked
in
Differential Equations
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set2
numerical-answers
differential-equations
0
answers
0
votes
GATE2018-2-4
If $y$ is the solution of the differential equation $y^3 \dfrac{dy}{dx}+x^3 = 0, \: y(0)=1,$ the value of $y(-1)$ is $-2$ $-1$ $0$ $1$
Arjun
asked
in
Differential Equations
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set2
differential-equations
0
answers
0
votes
GATE2018-2-3
Consider a function $u$ which depends on position $x$ and time $t$. The partial differential equation $\frac{\partial u}{\partial t} = \frac{\partial^2 u }{\partial x^2}$ is known as the Wave equation Heat equation Laplace's equation Elasticity equation
Arjun
asked
in
Differential Equations
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set2
differential-equations
partial-differential-equation
0
answers
0
votes
GATE2018-1-38
$F(s)$ is the Laplace transform of the function $f(t) =2t^2 e^{-t}$. $F(1)$ is _______ (correct to two decimal places).
Arjun
asked
in
Differential Equations
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set1
numerical-answers
differential-equation
laplace-transforms
0
answers
0
votes
GATE2018-1-37
An explicit forward Euler method is used to numerically integrate the differential equation $\dfrac{dy}{dt} = y$ using a time step of $0.1$. With the initial condition $y(0)=1$, the value of $y(1)$ computed by this method is ________ (correct to two decimal places)
Arjun
asked
in
Differential Equations
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set1
numerical-answers
differential-equations
euler-cauchy-equations
0
answers
0
votes
GATE2017 ME-2: 27
Consider the differential equation $3y" (x)+27 y (x)=0$ with initial conditions $y(0)=0$ and $y'(0)=2000$. The value of $y$ at $x=1$ is ________.
Arjun
asked
in
Differential Equations
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set2
numerical-answers
differential-equations
0
answers
0
votes
GATE2017 ME-2: 5
The Laplace transform of $te^{t}$ is $\dfrac{s}{(s+1)^{2}} \\$ $\dfrac{1}{(s-1)^{2}} \\$ $\dfrac{1}{(s+1)^{2}} \\$ $\dfrac{s}{(s-1)}$
Arjun
asked
in
Differential Equations
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set2
differential-equations
laplace-transforms
0
answers
0
votes
GATE2017 ME-1: 3
Consider the following partial differential equation for $u(x, y)$, with the constant $c > 1$: $\dfrac{\partial u}{\partial y}+c\dfrac{\partial u}{\partial x}=0$ Solution of this equation is $u(x, y) = f (x+cy)$ $u(x, y) = f (x-cy)$ $u(x, y) = f (cx+y)$ $u(x, y) = f (cx-y)$
Arjun
asked
in
Differential Equations
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set1
differential-equations
partial-differential-equation
0
answers
0
votes
GATE2017 ME-1: 4
The differential equation $\dfrac{d^{2}y}{dx^{2}}+16y=0$ for $y(x)$ with the two boundary conditions $\dfrac{dy}{dx}\bigg \vert _{x=0}=1$ and $\dfrac{dy}{dx}\bigg \vert_{x=\displaystyle \frac{\pi}{2}}=-1$ has. No solution. Exactly two solutions. Exactly one solution. Infinitely many solutions.
Arjun
asked
in
Differential Equations
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set1
differential-equations
initial-and-boundary-value-problems
0
answers
0
votes
GATE2016-3-3
Solutions of Laplace’s equation having continuous second-order partial derivatives are called biharmonic functions harmonic functions conjugate harmonic functions error functions
Arjun
asked
in
Differential Equations
Feb 24, 2017
by
Arjun
27.4k
points
gateme-2016-set3
differential-equations
laplace-transforms
0
answers
0
votes
GATE2016-2-3
Laplace transform of $\cos( \omega t)$ is $\dfrac{s}{s^2+\omega ^2} \\$ $\dfrac{\omega }{s^2+\omega ^2} \\$ $\dfrac{s}{s^2-\omega ^2} \\$ $\dfrac{\omega }{s^2-\omega ^2}$
Arjun
asked
in
Differential Equations
Feb 24, 2017
by
Arjun
27.4k
points
gateme-2016-set2
differential-equations
laplace-transforms
0
answers
0
votes
GATE2016-1-2
If $f(t)$ is a function defined for all $t \geq 0$, its Laplace transform $F(s)$ is defined as $\int_{0}^{\infty }e^{st}f(t)dt \\$ $\int_{0}^{\infty }e^{-st}f(t)dt \\$ $\int_{0}^{\infty }e^{ist}f(t)dt \\$ $\int_{0}^{\infty }e^{-ist}f(t)dt$
Arjun
asked
in
Differential Equations
Feb 24, 2017
by
Arjun
27.4k
points
gateme-2016-set1
differential-equations
laplace-transforms
0
answers
0
votes
GATE2015-3-44
Laplace transform of the function $f(t)$ is given by $F(s)=L\begin{bmatrix} f(t) \end{bmatrix}=\int_{0}^{\infty }f(t)e^{-st}dt$ . Laplace transform of the function shown below is given by $\displaystyle{\frac{1-e^{-2s}}{s}} \\$ $\displaystyle{\frac{1-e^{-s}}{2s}} \\$ $\displaystyle{\frac{2-2e^{-s}}{s}} \\$ $\displaystyle{\frac{1-2e^{-s}}{s}}$
Arjun
asked
in
Differential Equations
Feb 24, 2017
by
Arjun
27.4k
points
gateme-2015-set3
differential-equations
laplace-transforms
0
answers
0
votes
GATE2015-2-28
Consider the following differential equation: $\dfrac{dy}{dt}=-5y$; initial condition: $y=2$ at $t=0$. The value of $y$ at $t=3$ is $-5e^{-10}$ $2e^{-10}$ $2e^{-15}$ $-15e^{2}$
Arjun
asked
in
Differential Equations
Feb 24, 2017
by
Arjun
27.4k
points
gateme-2015-set2
differential-equations
0
answers
0
votes
GATE2015-2-4
The Laplace transform of $e^{i5t}$ where $i=\sqrt{-1}$, is $\dfrac{s-5i}{s^2-25} \\$ $\dfrac{s+5i}{s^2+25} \\$ $\dfrac{s+5i}{s^2-25} \\$ $\dfrac{s-5i}{s^2+25} $
Arjun
asked
in
Differential Equations
Feb 24, 2017
by
Arjun
27.4k
points
gateme-2015-set2
laplace-transforms
differential-equations
0
answers
0
votes
GATE2015-1-28
Find the solution of $\dfrac{d^2y}{dx^2}=Y$ which passes through the origin and the point $\left(\ln 2,\dfrac{3}{4}\right)$ $y=\dfrac{1}{2}e^x-e^{-x} $ $y=\dfrac{1}{2}(e^x+e^{-x}) $ $y=\dfrac{1}{2}(e^x-e^{-x}) $ $y=\dfrac{1}{2}e^x+e^{-x} $
Arjun
asked
in
Differential Equations
Feb 24, 2017
by
Arjun
27.4k
points
gateme-2015-set1
differential-equations
0
answers
0
votes
GATE2015-1-7
The Blasius equation related to boundary layer theory is a third-order linear partial differential equation third-order nonlinear partial differential equation second-order nonlinear ordinary differential equation third-order nonlinear ordinary differential equation
Arjun
asked
in
Differential Equations
Feb 24, 2017
by
Arjun
27.4k
points
gateme-2015-set1
differential-equations
boundary-value-problems
0
answers
0
votes
GATE Mechanical 2014 Set 4 | Question: 43
Consider the following statements regarding streamline(s): It is a continuous line such that the tangent at any point on it shows the velocity vector at that point There is no flow across streamlines $\dfrac{dx}{u}=\dfrac{dy}{v}=\dfrac{dz}{w}$ is the differential equation of a ... $(ii), (iii), (iv)$ $(i), (iii), (iv)$ $(i), (ii), (iii)$
Arjun
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Differential Equations
Feb 19, 2017
by
Arjun
27.4k
points
gateme-2014-set4
differential-equations
0
answers
0
votes
GATE Mechanical 2014 Set 4 | Question: 5
Laplace transform of $\cos(\omega t)$ is $\dfrac{s}{s^2+\omega ^2}$. The Laplace transform of $e^{-2t} \cos(4t)$ is $\dfrac{s-2}{(s-2)^2+16} \\$ $\dfrac{s+2}{(s-2)^2+16} \\$ $\dfrac{s-2}{(s+2)^2+16} \\$ $\dfrac{s+2}{(s+2)^2+16}$
Arjun
asked
in
Differential Equations
Feb 19, 2017
by
Arjun
27.4k
points
gateme-2014-set4
differential-equations
laplace-transforms
0
answers
0
votes
GATE Mechanical 2014 Set 4 | Question: 3
The solution of the initial value problem $\dfrac{dy}{dx}=-2xy$ ; $y(0)=2$ is $1+e^{{-x}^2}$ $2e^{{-x}^2}$ $1+e^{{x}^2}$ $2e^{{x}^2}$
Arjun
asked
in
Differential Equations
Feb 19, 2017
by
Arjun
27.4k
points
gateme-2014-set4
differential-equations
0
answers
0
votes
GATE Mechanical 2014 Set 3 | Question: 27
Consider two solutions $x(t)=x_1(t)$ and $x(t)=x_2(t)$ of the differential equation $\dfrac{d^2x(t)}{dt^2}+x(t)=0, \: t>0$ such that $x_1(0)=1, \dfrac{dx_1(t)}{dt} \bigg \vert_{t=0}=0$, $x_2(0)=0, \dfrac{dx_2(t)}{dt}\bigg \vert _{t=0}=1$. The ... $t=\pi /2$ is $1$ $-1$ $0$ $\pi /2$
Arjun
asked
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Differential Equations
Feb 19, 2017
by
Arjun
27.4k
points
gateme-2014-set3
differential-equations
0
answers
0
votes
GATE Mechanical 2014 Set 2 | Question: 27
The general solution of the differential equation $\dfrac{dy}{dx}=\cos(x+y)$, with $c$ as a constant, is $y+\sin \left (x+y \right )=x+c \\$ $\tan \left (\dfrac{x+y}{2} \right)=y+c \\$ $\cos \left (\dfrac{x+y}{2} \right )=x+c \\$ $\tan \left (\dfrac{x+y}{2} \right )=x+c$
Arjun
asked
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Differential Equations
Feb 19, 2017
by
Arjun
27.4k
points
gateme-2014-set2
differential-equations
0
answers
0
votes
GATE ME 2013 | Question: 46
The solution to the differential equation $\dfrac{d^2u}{dx^2}-k\dfrac{du}{dx}=0$ where $k$ is a constant, subjected to the boundary conditions $u(0)$ = $0$ and $u(L)$ = $U$, is $u=U\dfrac{x}{L} $ $u=U\left(\dfrac{1-e^{kx}}{1-e^{kL}}\right) $ $u=U\left(\dfrac{1-e^{-kx}}{1-e^{-kL}}\right)$ $u=U\left(\dfrac{1+e^{kx}}{1+e^{kL}}\right)$
piyag476
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Differential Equations
Feb 19, 2017
by
piyag476
1.4k
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gateme-2013
differential-equations
initial-and-boundary-value-problems
0
answers
0
votes
GATE ME 2013 | Question: 27
The function $f(t)$ satisfies the differential equation $\dfrac{d^2f}{dt^2}+f=0$ and the auxiliary conditions, $f(0)=0$, $\dfrac{d(f)}{d(t)}(0)=4$. The Laplace transform of $f(t)$is given by $\dfrac{2}{s+1} \\$ $\dfrac{4}{s+1} \\$ $\dfrac{4}{s^2+1} \\$ $\dfrac{2}{s^4+1}$
piyag476
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Differential Equations
Feb 19, 2017
by
piyag476
1.4k
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gateme-2013
differential-equation
laplace-transforms
0
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0
votes
GATE ME 2013 | Question: 1
The partial differential equation $\dfrac{\partial u }{\partial t}+u\dfrac{\partial u}{\partial x}=\dfrac{\partial^2 u}{\partial x^2}$ is a linear equation of order $2$ non-linear equation of order $1$ linear equation of order $1$ non-linear equation of order $2$
piyag476
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Differential Equations
Feb 19, 2017
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piyag476
1.4k
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gateme-2013
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