GATE Overflow for GATE BT - Recent activity in Differential Equations

Edited: GATE BT 2026 | Question: 48

Tue, 17 Mar 2026 07:30:34 +0000

From the following plot of $\dfrac{d y}{d x}$ versus $x$ and if $y(2)=5$, the value of $y(3)$ is $\_\_\_\_$. (rounded off to one decimal place)

Edited: GATE BT 2026 | Question: 44

Tue, 17 Mar 2026 07:17:52 +0000

A heat exchanger during operation in a bioprocess has a steady temperature of $90^{\circ} \mathrm{C}$. After completion of its operation, it was shut down and it was observed that the rate of decrease of temperature at any time was directly proportional to the difference $T(t)-30^{\circ} \mathrm{C}$, where $T(t)$ denotes temperature at time $t$. It was observed that it took $30$ min for the temperature to drop to $70^{\circ} \mathrm{C}$. The temperature after $51.5$ min will be $\_\_\_\_$ ${ }^{\circ} \mathrm{C}$. (rounded off to the nearest integer)

Edited: GATE BT 2026 | Question: 17

Tue, 17 Mar 2026 06:56:48 +0000

The equation $\dfrac{d^{2} y}{d x^{2}}-y=0$ has a solution of the form $y=e^{A x}$. The value(s) of $A$ satisfying this is/are:

$0$
$1$
$-1$
$-\infty$

Answered: GATE BT 2025 | Question: 3

Wed, 04 Feb 2026 18:56:49 +0000

Edited: GATE BT 2025 | Question: 35

Mon, 02 Jun 2025 13:45:01 +0000

Let $m$ and $n$ be fixed real numbers. If the function $y(t)=C_{1} e^{t}+C_{2} e^{-t}$ is a solution of
$$ \frac{d^{2} y}{d t^{2}}+m \frac{d y}{d t}+n y=0$$
for any constants $C_{1}$ and $C_{2}$, then $m+n$ is equal to

$-2$
$-1$
$0$
$1$

Recategorized: GATE BT 2024 | Question: 20

Mon, 21 Apr 2025 15:49:42 +0000

The solution of the differential equation $\frac{d y}{d x}=y+e^{-x}$ that satisfies $y(0)=-\frac{1}{2}$ is $\_\_\_\_\_\_\_$.

$-\frac{1}{2} e^{-\frac{x}{2}}$
$-\frac{1}{2} e^{x}$
$-\frac{1}{2} e^{-x}$
$-\frac{1}{2} e^{\frac{x}{2}}$

Recategorized: GATE BT 2024 | Question: 54

Mon, 21 Apr 2025 15:48:10 +0000

Let $y(x)=x^2 \ln x$ for $x>0$, be a solution of $x^2 \frac{d^2 y}{d x^2}+4 y=\alpha x \frac{d y}{d x}$. Then the value of $\alpha$ is $\_\_\_\_\_\_\_\_\_$.

Recategorized: GATE BT 2022 | Question: 1

Mon, 21 Apr 2025 15:43:25 +0000

What is the order of the differential equation given below?

$\dfrac{d^{2}y}{dx^{2}} – 6x = 3x^{4} – 2x^{3} + 2$

Recategorized: GATE2020: 25

Mon, 21 Apr 2025 15:36:19 +0000

A variable $Y$ is a function of $t$. Given that $Y\left ( t=0 \right )=1$ and $Y\left ( t=1 \right )=2,\dfrac{dY}{dt}$ in the interval $t=\left [ 0,1 \right ]$ can be approximated as _________________.

Recategorized: GATE2020: 18

Mon, 21 Apr 2025 15:36:19 +0000

Given that $Z=X^{2}+Y^{2}$, the value of $\dfrac{\partial Z}{\partial X}$ for $X=1$ and $Y=0$ is ________________ (answer is an integer).

Retagged: GATE2019: 38

Mon, 21 Apr 2025 15:35:31 +0000

What is the solution of the differential equation $\dfrac{\mathrm{dy} }{\mathrm{d} x}=\dfrac{x}{y}$, with the initial condition, at $x=0, y=1?$

$x^{2}=y^{2}+1$
$y^{2}=x^{2}+1$
$y^{2}=2x^{2}+1$
$x^{2}-y^{2}=0$

Recategorized: GATE2019: 39

Mon, 21 Apr 2025 15:33:47 +0000

The Laplace transform of the function $f\left ( t \right )=t^{2}+2t+1$ is

$\dfrac{1}{s^{3}}+\dfrac{3}{s^{2}}+\dfrac{1}{s} \\$
$\dfrac{4}{s^{3}}+\dfrac{4}{s^{2}}+\dfrac{1}{s} \\$
$\dfrac{1}{s^{3}}+\dfrac{2}{s^{2}}+\dfrac{1}{s} \\$
$\dfrac{2}{s^{3}}+\dfrac{2}{s^{2}}+\dfrac{1}{s}$

Recategorized: GATE2019: 12

Mon, 21 Apr 2025 15:33:47 +0000

Which one of the following equations represents a one-dimensional wave equation?

$\dfrac{\partial u}{\partial t}=C^{2}\dfrac{\partial ^{2}u}{\partial x^{2}} \\$
$\dfrac{\partial ^{2}u}{\partial t^{2}}=C^{2}\dfrac{\partial ^{2}u}{\partial x^{2}} \\$
$\dfrac{\partial ^{2}u}{\partial t^{2}}=C^{2}\dfrac{\partial u}{\partial x} \\$
$\dfrac{\partial ^{2}u}{\partial t^{2}}+\dfrac{\partial ^{2}u}{\partial x^{2}}=0$

Recategorized: GATE2014-26

Mon, 21 Apr 2025 15:25:00 +0000

The concentration profile of a chemical at a location $x$ and time $t$, denoted by $c(x,t)$, changes as per the following equation,

$$c(x,t)=\frac{c_0}{\sqrt{2\pi Dt}}\exp[-\frac{x^2}{2Dt}]$$

where $D$ and $c_0$ are assumed to be constant. Which of the following is correct?

$\frac{\partial c}{\partial t}=D\frac{\partial^2 c}{\partial x^2}$
$\frac{\partial c}{\partial t}=\frac{D}{2}\frac{\partial^2 c}{\partial x^2}$
$\frac{\partial^2 c}{\partial t^2}=D\frac{\partial^2 c}{\partial x^2}$
$\frac{\partial^2 c}{\partial t^2}=\frac{D}{2}\frac{\partial^2 c}{\partial x^2}$

Retagged: GATE BT 2013 | Question: 41

Mon, 21 Apr 2025 15:23:06 +0000

The solution to $\frac {dy}{dx}+y \cot x=\csc x$ is

$y=(c+x)\cot x$
$y=(c+x)\csc x$
$y=(c+x)\csc x\cot x$
$y=(c+x)\frac{\csc x}{\cot x}$

Recategorized: GATE BT 2017 | Question: 16

Thu, 18 Mar 2021 17:06:41 +0000

For $y=f(x)$, if $\dfrac{d^2y}{dx^2}=0$, $\dfrac{dy}{dx}=0$ at $x=0$, and $y=1$ at $x=1$, the value of $y$ at $x=2$ is __________

Recategorized: GATE2016-53

Thu, 18 Mar 2021 17:01:15 +0000

$\dfrac{d^2y}{dx^2}-y=0$. The conditions for this second order homogeneous differential equation are $y(0)=1$ and $\dfrac{dy}{dx}=3$ at $x=0$. The value of $y$ when $x = 2$ is __________

Recategorized: GATE2016-24

Thu, 18 Mar 2021 17:00:23 +0000

The Laplace transform $F(s)$ of the function $f(t) = \cos (at)$, where $a$ is constant, _________

$\dfrac{s^2}{s^2+a^2} \\$
$\dfrac{a}{s^2+a^2} \\$
$\dfrac{s}{s^2+a^2} \\$
$\dfrac{s}{s^2-a^2} $

Retagged: GATE BT 2013 | Question: 39

Thu, 18 Mar 2021 16:12:10 +0000

The Laplace transform of $f(t) = 2t + 6$ is

$\frac{1}{s}+\frac{2}{s^2}$
$\frac{3}{s}-\frac{6}{s^2}$
$\frac{6}{s}+\frac{2}{s^2}$
$-\frac{6}{s}+\frac{2}{s^2}$