GATE Overflow for GATE ESE - Recent questions and answers in Differential Equations

GATE ES 2026 | Question: 26

Mon, 23 Feb 2026 13:15:00 +0000

A first-order ordinary differential equation is given as follows:

$$\dfrac{d y}{d x}+x^{2} y=0$$

Which ONE of the following options CORRECTLY represents the characteristics of this equation?

Linear, homogeneous, and exact
Nonlinear, nonhomogeneous, and exact
Linear, homogeneous, and non-exact
Nonlinear, nonhomogeneous, and non-exact

GATE ES 2025 | Question: 2

Thu, 06 Mar 2025 16:22:54 +0000

For the ordinary differential equation $\frac{d^{2} y}{d x^{2}}+4 y=0$, the general solution is

$y=c_{1} \cos 2 x+c_{2} \sin 2 x$
$y=c_{1} \cosh 2 x+c_{2} \sinh 2 x$
$y=c_{1} e^{2 x}+c_{2} e^{-2 x}$
$y=c_{1} e^{2 x} \cos 2 x+c_{2} e^{-2 x} \sin 2 x$

GATE ES 2025 | Question: 40

Thu, 06 Mar 2025 16:21:13 +0000

A tank has inflow, outflow and stirring mechanism. Initially, the tank holds $500 \: L$ of a brine solution of concentration $200 \mathrm{~g} / \mathrm{L}$. At $t=0$, an inflow of another brine solution of concentration $100 \mathrm{~g} / \mathrm{L}$ starts entering the tank at the rate of $15 \mathrm{~L} /$ minute. At the same time the outflow of thoroughly stirred mixture also takes place at the same rate so that the volume of brine in the tank remains constant. The brine concentration $C(\mathrm{~g} / \mathrm{L})$ in the tank at any time $t$ (minute) can be expressed by the following differential equation
$$\frac{d C}{d t}+0.03 C=3$$

The brine concentration in the tank at $t=1.5$ hour is $\_\_\_\_\_\_ \: \mathrm{g/L}$. (rounded off to two decimal places)

GATE ES 2024 | Question: 23

Mon, 19 Feb 2024 10:24:10 +0000

Consider the equation $\frac{d y}{d x}-x^{2}+e^{x}=0$; with $y=1$ at $x=0$.
The value of $y$ at $x=1$ is ________ $\text{(rounded off to 2 decimal places)}$.

Take the value of $e$ $\text{(base of natural logarithm)}$ as $2.7$.

GATE ES 2024 | Question: 27

Mon, 19 Feb 2024 10:24:09 +0000

Which one is the solution $y(x)$ for the following ordinary differential equation and the specified boundary conditions?

\[
\frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+2 y=2 e^{-x} ; y(0)=2 ; \quad\left(\frac{d y}{d x}\right)_{x=0}=1
\]

$y(x)=\frac{1}{3} e^{-x}-2 e^{x}-\frac{1}{3} e^{2 x}$
$y(x)=\frac{1}{3} e^{x}+2 e^{x}-\frac{1}{3} e^{2 x}$
$y(x)=\frac{1}{3} e^{-x}+2 e^{-x}-\frac{1}{3} e^{2 x}$
$y(x)=\frac{1}{3} e^{-x}+2 e^{x}-\frac{1}{3} e^{2 x}$

GATE ES 2023 | Question: 1

Tue, 23 May 2023 03:12:50 +0000

Given are two ordinary differential equations

$\mathrm{P}: \frac{d y}{d x}+x=x \sin y$

$\mathrm{Q}: \frac{d y}{d x}+x y=e^{x} y$

The correct choice is

$\mathrm{P}$ is linear; $\mathrm{Q}$ is nonlinear
$\mathrm{P}$ is nonlinear; $\mathrm{Q}$ is linear
Both $\mathrm{P}$ and $\mathrm{Q}$ are linear
Both $\mathrm{P}$ and $\mathrm{Q}$ are nonlinear

GATE ES 2023 | Question: 37

Tue, 23 May 2023 03:12:01 +0000

Second order ordinary differential equation $\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-2 y=0$ has values $y=2$ and $\frac{d y}{d x}=1$ at $x=0$. The value of y at $x=1$ is _____________ $\text { off to three decimal places).}$

GATE ES 2021 | Question: 20

Tue, 14 Mar 2023 03:44:29 +0000

The ordinary differential equation
\[
\frac{d y}{d x}=x^{2} y
\]
has $y$ as the dependent variable and $x$ as the independent variable. Which of the following classification(s) is/are applicable to the equation?

Linear
Non-linear
First order
Second order

GATE ES 2021 | Question: 42

Tue, 14 Mar 2023 03:43:58 +0000

Consider a function $y=f(x)$ which satisfies the following equation:
$\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}=0$
As $x \rightarrow-\infty, y=1$, and at $x=0, y=2$.

The value of $\frac{d y}{d x}$ at $x=0$ is _________________ $\text{(answer in integer)}$

GATE ES 2022 | Question: 26

Fri, 17 Feb 2023 07:05:52 +0000

Given, $y=f(x) ; \frac{d^{2} y}{d x^{2}}+4 y=0 ; y(0)=0 ; \frac{d y}{d x}(0)=1$. The problem is a/an

initial value problem having solution $\mathrm{y}=x$
boundary value problem having solution $\mathrm{y}=x$
initial value problem having solution $\mathrm{y}=\frac{1}{2} \sin 2 x$
boundary value problem having solution $\mathrm{y}=\frac{1}{2} \sin 2 x$