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Recent questions and answers in Differential Equations
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TIFR ECE 2021 | Question: 4
The first-order differential equation $\frac{d y(t)}{d t}+2 y(t)=x(t)$ describes a particular continuous-time system initially at rest at origin i.e., $x(0)=0$. Consider the following statements? System is memoryless. System is causal. System is stable. Which of the ... correct. All $(1), (2)$ and $(3)$ are correct. Only $(2)$ and $(3)$ are correct. None of the above
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GATE ECE 2009 | Question: 1
The order of the differential equation $\dfrac{d^{2} y}{d t^{2}}+\left(\dfrac{d y}{d t}\right)^{3}+y^{4}=e^{-t} \quad$ is $1$ $2$ $3$ $4$
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GATE ECE 2010 | Question: 3
A function $n(x)$ satisfies the differential equation $\frac{d^{2} n(x)}{d x^{2}}-\frac{n(x)}{L^{2}}=0$ where $L$ is a constant. The boundary conditions are: $n(0)=K$ and $n(\infty)=0$. The solution to this equation is $n(x)=K \exp (x / L)$ $n(x)=K \exp (-x / \sqrt{L})$ $n(x)=K^{2} \exp (-x / L)$ $n(x)=K \exp (-x / L)$
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GATE ECE 2010 | Question: 30
Consider a differential equation $\dfrac{d y(x)}{d x}-y(x)=x$ with the initial condition $y(0)=0$. Using Euler's first order method with a step size of $0.1$, the value of $y(0.3)$ is $0.01$ $0.031$ $0.0631$ $0.1$
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GATE ECE 2011 | Question: 25
The solution of the differential equation $\frac{d y}{d x}=k y, y(0)=c$ is $x=c e^{-k y}$ $x=k e^{c y}$ $y=c e^{k x}$ $y=c e^{-k x}$
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GATE ECE 2021 | Question: 2
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 $\left ( 1-x^{2} \right )^{-3/4}$ $\left ( 1-x^{2} \right )^{-1/4}$ $\left ( 1-x^{2} \right )^{-3/2}$ $\left ( 1-x^{2} \right )^{-1/2}$
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GATE ECE 2020 | Question: 4
The general solution of $\dfrac{\mathrm{d^{2}} y}{\mathrm{d} x^{2}}-6\dfrac{\mathrm{d} y}{\mathrm{d} x}+9y=0$ is $y=C_{1}e^{3x}+C_{2}e^{-3x}$ $y=(C_{1}+C_{2}x)e^{-3x}$ $y=(C_{1}+C_{2}x)e^{3x}$ $y=C_{1}e^{3x}$
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GATE ECE 2020 | Question: 27
Which one of the following options contains two solutions of the differential equation $\dfrac{\mathrm{d} y}{\mathrm{d} x}=\left ( y-1 \right )x?$ $\ln\mid y-1 \mid=0.5x^{2}+C$ and $y=1$ $\ln\mid y-1 \mid=2x^{2}+C$ and $y=1$ $\ln\mid y-1 \mid=0.5x^{2}+C$ and $y=-1$ $\ln\mid y-1 \mid=2x^{2}+C$ and $y=-1$
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GATE ECE 2019 | Question: 2
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 Parabolas and Circles Circles and Hyperbolas Hyperbolas and Circles Hyperbolas and Parabolas
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GATE ECE 2019 | Question: 43
Consider the homogenous 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 $y(1)=1 \quad \text{and} \quad y(2)=14$ the value of $y(1.5),$ rounded off to two decimal places, is________.
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GATE ECE 2016 Set 3 | Question: 26
The particular solution of the initial value problem given below is $\frac{d^2y}{dx^2}+12\frac{dy}{dx}+36y=0\hspace{0.3cm} \text{ with } \hspace{0.3cm}y(0)=3\hspace{0.3cm} \text{ and }\hspace{0.3cm} \frac{dy}{dx} \bigg| _{x=0} =-36$ $(3-18x)e^{-6x}$ $(3+25x)e^{-6x}$ $(3+20x)e^{-6x}$ $(3-12x)e^{-6x}$
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GATE ECE 2016 Set 2 | Question: 19
The response of the system $G(s)=\frac{s-2}{(s+1)(s+3)}$ to the unit step input $u(t)$ is $y(t)$. The value of $\frac{dy}{dt}$ at $t=0^{+}$ is _________
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GATE ECE 2016 Set 2 | Question: 26
The ordinary differential equation $\frac{dx}{dt}=-3x+2, \text{ with }x(0) = 1$ 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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14
GATE ECE 2015 Set 3 | Question: 28
Consider the differential equation $\dfrac{\mathrm{d^{2}}x(t) }{\mathrm{d} t^{2}} +3\frac{\mathrm{d}x(t)}{\mathrm{d} t} + 2x(t) = 0. $ Given $x(0) = 20$ and $x(1) = 10/e,$ where $e = 2.718,$ the value of $x(2)$ is ________.
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15
GATE ECE 2015 Set 2 | Question: 4
The general solution of the differential equation $\dfrac{\mathrm{d} y}{\mathrm{d} x} = \dfrac{1+\cos 2y}{1-\cos 2x}$ is $ \tan y – \cot x = c\:\text{(c is a constant)}$ $\tan x – \cot y = c\:\text{(c is a constant)}$ $\tan y + \cot x = c\:\text{(c is a constant)}$ $\tan x + \cot y = c\:\text{(c is a constant)}$
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GATE ECE 2015 Set 2 | Question: 26
Consider the differential equation $\dfrac{\mathrm{d} x }{\mathrm{d} t} = 10 – 0.2x$ with initial condition $x(0) = 1$. The response $x(t)$ for $t>0$ is $2-e^{-0.2t}$ $2-e^{0.2t}$ $50-49e^{-0.2t}$ $50-49e^{0.2t}$
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17
GATE ECE 2015 Set 1 | Question: 25
The solution of the differential equation $\frac{d^2y}{dt^2} + 2 \frac{dy}{dt}+y=0$ with $y(0)=y’(0)=1$ is $(2-t)e^t$ $(1+2t)e^{-t}$ $(2+t)e^{-t}$ $(1-2t)e^t$
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18
GATE ECE 2014 Set 4 | Question: 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 $ae^{-t}$ $ae^{-t} + bte^{-t}$ $ae^t+bte^{-t}$ $ae^{-2t}$
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19
GATE ECE 2014 Set 4 | Question: 26
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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20
GATE ECE 2014 Set 3 | Question: 2
Which $ONE$ of the following is a linear non-homogeneous differential equation, where $x$ and $y$ are the independent and dependent variables respectively? $\frac{dy}{dx}+xy= e^{-x}$ $\frac{dy}{dx}+xy= 0$ $\frac{dy}{dx}+xy= e^{-y}$ $\frac{dy}{dx}+ e^{-y}= 0$
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21
GATE ECE 2014 Set 3 | Question: 5
If $z= xy \text{ ln} (xy)$, then $x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}= 0 \\$ $y\frac{\partial z}{\partial x}= x\frac{\partial z}{\partial y} \\$ $x\frac{\partial z}{\partial x}= y\frac{\partial z}{\partial y} \\$ $y\frac{\partial z}{\partial x}+x\frac{\partial z}{\partial y}= 0$
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GATE ECE 2014 Set 2 | Question: 5
If the characteristic equation of the differential equation $\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}+2\alpha \frac{\mathrm{d}y}{\mathrm{d} x}+y= 0$ has two equal roots, then the value of $\alpha$ are $\pm 1$ $0,0$ $\pm j$ $\pm 1/2$
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23
GATE ECE 2014 Set 1 | Question: 45
A system is described by the following differential equation, where $u(t)$ is the input to the system and $y(t)$ is the output of the system. $y(t) + 5y(t) = u(t)$ When $y(0) = 1$ and $u(t)$ is a unit step function, $y(t)$ is $0.2 + 0.8e^{-5t}$ $0.2 - 0.2e^{-5t}$ $0.8 + 0.2e^{-5t}$ $0.8 - 0.8e^{-5t}$
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GATE ECE 2014 Set 1 | Question: 48
For the following feedback system $G(s) = \dfrac{1}{(s+1)(s+2)}.$ The $2\%$-settling time of the step response is required to be less than $2$ seconds. Which one of the following compensators $C(s)$ achieves this? $3\bigg(\dfrac{1}{s+5}\bigg) \\$ $5\bigg(\dfrac{0.03}{s} + 1\bigg) \\$ $2(s+4) \\$ $4\bigg(\dfrac{s+8}{s+3}\bigg)$
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25
GATE ECE 2013 | Question: 37
A system described by a linear, constant coefficient, ordinary, first order differential equation has an exact solution given by $y(t)$ for $t>0,$ when the forcing function is $x(t)$ and the initial condition is $y(0).$ If one wishes to modify the ... forcing function to $j\sqrt{2}x(t)$ change the initial condition to $−2y(0)$ and the forcing function to $−2x(t)$
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GATE ECE 2013 | Question: 36
A system is described by the differential equation $\dfrac{\mathrm{d}^{2} y}{\mathrm{d} x} + 5\dfrac{\mathrm{d}y }{\mathrm{d} x} + 6y(t) = x(t).$ Let $x(t)$ be a rectangular pulse given by $x(t) = \begin{cases} 1&0<t<2 \\ 0&\text{otherwise} \end{cases}$ ... $\frac{e^{-2s}}{(s+2)(s+3)} \\$ $\frac{1-e^{-2s}}{s(s+2)(s+3)} $
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GATE ECE 2012 | Question: 34
Consider the differential equation $\frac{d^2y(t)}{dt^2}+2\frac{dy(t)}{dt}+y(t)=\delta(t)$ with $y(t)\big|_{t=0^-}=-2$ and $\frac{dy}{dt}\big|_{t=0^-}=0$. The numerical value of $\frac{dy}{dt}\big|_{t=0^+}$ is $-2$ $-1$ $0$ $1$
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GATE ECE 2012 | Question: 12
With initial condition $x(1)=0.5$, the solution of the differential equation, $t\frac{dx}{dt}+x=t$ is $x=t-\frac{1}{2}$ $x=t^2-\frac{1}{2}$ $x=\frac{t^2}{2}$ $x=\frac{t}{2}$
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29
GATE ECE 2018 | Question: 50
The position of a particle $y\left ( t \right )$ is described by the differential equation: $\frac{d^{2}y}{dt^{2}}=-\frac{dy}{dt}-\frac{5y}{4}.$ The initial conditions are $y\left ( 0 \right )=1$ and $\frac{dy}{dt}\mid_{t=0}=0$. The position (accurate to two decimal places) of the particle at $t=\pi$ is _________.
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GATE ECE 2018 | Question: 34
A curve passes through the point $\left ( x=1,y=0 \right )$ and satisfies the differential equation $\dfrac{\mathrm{dy} }{\mathrm{d} x}=\dfrac{x^{2}+y^{2}}{2y}+\dfrac{y}{x}.$ The equation that describes the curve is $\ln\left (1+\dfrac{y^{2}}{x^{2}}\right)=x-1$ ... $\ln\left (1+\dfrac{y}{x}\right)=x-1$ $\dfrac{1}{2}\ln\left (1+\dfrac{y}{x}\right)=x-1$
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31
GATE ECE 2018 | Question: 12
Let $f\left ( x,y \right )=\dfrac{ax^{2}+by^{2}}{xy},$ where $a$ and $b$ are constants. If $\dfrac{\partial f}{\partial x}=\dfrac{\partial f}{\partial y}$ at $x = 1$ and $y = 2$, then the relation between $a$ and $b$ is $a=\dfrac{b}{4}$ $a=\dfrac{b}{2}$ $a=2b$ $a=4b$
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GATE ECE 2018 | Question: 4
Let the input be $u$ and the output be $y$ ... $y=au+b,b\neq 0$ $y=au$
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GATE ECE 2017 Set 2 | Question: 2
The general solution of the differential equation $\frac{d^2y}{dx^2}+2\frac{dy}{dx}-5y=0$ in terms of arbitrary constants $K_1$ and $K_2$ is $K_1e^{(-1+\sqrt{6})x}+K_2e^{(-1-\sqrt{6})x}$ $K_1e^{(-1+\sqrt{8})x}+K_2e^{(-1-\sqrt{8})x}$ $K_1e^{(-2+\sqrt{6})x}+K_2e^{(-2-\sqrt{6})x}$ $K_1e^{(-2+\sqrt{8})x}+K_2e^{(-2-\sqrt{8})x}$
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GATE ECE 2017 Set 1 | Question: 29
Which one of the following is the general solution of the first order differential equation $\frac{dy}{dx}=(x+y-1)^{2},$ where $x,y$ are real? $y=1+x+\tan^{-1}(x+c)$, where $c$ is a constant $y=1+x+\tan(x+c)$, where $c$ is a constant $y=1-x+\tan^{-1}(x+c)$, where $c$ is a constant $y=1-x+\tan(x+c)$, where $c$ is a constant
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Recent questions and answers in Differential Equations