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Recent questions and answers in Differential Equations

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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...
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GATE ECE 2024 | Question: 1
The general form of the complementary function of a differential equation is given by $y(t)=(A t+B) e^{-2 t}$, where $A$ and $B$ are real constants determined by the init...
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GATE ECE 2025 | Question: 22
The function $y(t)$ satisfies$$t^{2} y^{\prime \prime}(t)-2 t y^{\prime}(t)+2 y(t)=0,$$where $y^{\prime}(t)$ and $y^{\prime \prime}(t)$ denote the first and second deriva...
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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 ...
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GATE ECE 2004 | Question: 36
A system described by the following differential equation $\frac{d^{2} y}{d t^{2}}+3 \frac{d y}{d t}+2 y=x(t)$ is initially at rest. For input $x(t)=2 u(t)$, the output $...
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GATE ECE 2005 | Question: 1
The following differential equation has\[3 \frac{d^{2} y}{d t^{2}}+4\left(\frac{d y}{d t}\right)^{3}+y^{2}+2=x\]degree $=2$, order $=1$degree $=3$, order $=2$degree $=4$,...
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GATE ECE 2005 | Question: 4
A solution of the following differential equation is given by\[\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0\]$y=e^{2 x}+e^{-3 x}$$y=e^{2 x}+e^{3 x}$$y=e^{-2 x}+e^{3 x}...
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GATE ECE 1995 | Question 1.15
The solution of $X=A(t) X(t)$, is$e^{\mathrm{At}} \cdot \mathrm{X}_{0}$$e^{\int_{t_{a}}^{\prime} A(\tau) d \tau}. X_{0}$$\left[I+\int_{t_{a}}^{t} \mathrm{~A}(\tau) d \tau...
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GATE ECE 2006 | Question: 5
A solution for the differential equation$ \qquad \dot{x}(t)+2 x(t)=\delta(t)$with initial condition $x(0-)=0$ is$e^{-2 t} \; u(t)$$e^{2 t} \; u(t)$$e^{-t} \; u(t)$$e^{t} ...
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GATE ECE 2006 | Question: 27
For the differential equation $\dfrac{d^2 y}{d x^2}+k^2 y=0$, the boundary conditions are$y=0$ for $x=0$, and$y=0$ for $x=a$The form of non-zero solutions of $y$ (where $...
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GATE ECE 1994 | Question 4.1
(A) $a_{1} \frac{d^{2} y}{d x^{2}}+a_{2} y \frac{d y}{d x}+a_{3} y=a_{4}$(B) $a_{1} \frac{d^{3} y}{d x^{3}}+a_{2} y=a_{3}$(C) $a_{1} \frac{d^{2} y}{d x^{2}}+a_{2} x \frac...
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GATE ECE 2007 | Question: 24
The solution of the differential equation $k^{2} \dfrac{d^{2} y}{d x^{2}}=y-y_{2}$ under the boundary conditions (i) $y=y_{1}$ at $x=0$ and (ii) $y=y_{2}$ at $x=\infty$, ...
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GATE ECE 2008 | Question: 6
Which of the following is a solution to the differential equation $\dfrac{d x(t)}{d t}+3 x(t)=0?$ $x(t)=3 e^{-t}$$x(t)=2 e^{-3 t}$$x(t)=-\frac{3}{2} t^{2}$$x(t)=3 t^{2}$
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GATE ECE 1993 | Question 1.2
The differential equation, $\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+\sin$ $y=0$, islinearnon-linearhomogeneousof degree two
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GATE ECE 1993 | Question 2.9
The differential equation $y+y=0$ is subjected to the boundary conditions\[y(0)+0 y(\lambda)=0\]In order that the equation has non-trivial solution (s), the general value...
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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$$4$
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GATE ECE 2009 | Question: 25
Match each differential equation in $\text{Group I}$ to its family of solution curves from $\text{Group II.}$$$\begin{array}{llll} & \textbf{Group I} & & \textbf{Group II...
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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...
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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 o...
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GATE ECE 2010 | Question: 41
A continuous time $\text{LTI}$ system is described by\[ \frac{d^{2} y(t)}{d t^{2}}+4 \frac{d y(t)}{d t}+3 y(t)=2 \frac{d x(t)}{d t}+4 x(t) \]Assuming zero initial conditi...
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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}$$y=c e^{-k x}$
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GATE ECE 2022 | Question: 15
Consider the following partial differential equation $\text{(PDE)}$$$a \frac{ \partial^{2} f(x ,y)}{\partial x^{2}} + b \frac{ \partial^{2} f(x ,y)}{\partial y^{2}} = f(x...
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GATE ECE 2022 | Question: 20
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 s...
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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...
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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?$$\ln\mid y-1 \mid=0.5x^{...
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GATE ECE 2019 | Question: 2
The families of curves represented by the solution of the equation$$\frac{dy}{dx}=\: – \left(\frac{x}{y} \right)^n$$for $n=-1$ and $n= +1,$ respectively, areParabolas and...
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GATE ECE 2019 | Question: 43
Consider the homogenous ordinary differential equation$$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 ...
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homogenous diff. eq. with I.F. 1/Mx+Ny
please solve$\left ( 2x-y \right )e^{\frac{y}{x}}dx+\left ( y+xe^{\frac{y}{x}} \right )dy$
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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...
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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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