GATE Overflow for GATE XE - Recent questions in Ordinary Differential Equations

GATE XE 2026 | Question: 3

Tue, 24 Feb 2026 15:50:29 +0000

Let $W(x)$ denote the Wronskian of two linearly independent solutions $y_{1}(x)$ and $y_{2}(x)$ of the differential equation

$$(x-1) \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+x e^{x} y=0, \quad x>1 .$$

If $W(2)=2$, then the value of $W(5)$ is

$\dfrac{1}{8}$
$\dfrac{1}{5}$
$5$
$8$

GATE XE 2026 | Question: 11

Tue, 24 Feb 2026 15:49:46 +0000

The initial value problem

$$\begin{array}{l}
\dfrac{d u}{d t}=u^{2}+t^{2}, \quad t \geq 0 \\
\text { with } u(0)=1
\end{array}$$

is solved by using the explicit Euler method with step size $h=0.2$. Then, the value of $u(0.4)$ (rounded off upto two decimal places) is $\_\_\_\_$

GATE XE 2025 | Question: 16

Sun, 04 May 2025 19:08:35 +0000

Let $y(x)$ be the solution of the initial value problem,

$$\begin{array}{c}
x^{2} y^{\prime \prime}+x y^{\prime}-y=0, \quad x>0, \\
y(1)=0, \quad y^{\prime}(1)=2 .
\end{array}$$

Then the value of $y^{\prime}\left(\frac{1}{2}\right)$ is equal to (Answer in integer) $\_\_\_\_\_\_\_$

GATE XE 2024 | Question: 2

Sun, 21 Jul 2024 16:42:44 +0000

Let $y$ be a non-zero quadratic polynomial satisfying the differential equation
\[
\left(2+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-k y=0,
\]
where $k$ is a real constant. If $y(1)=1$, then the value of the integral
\[
\int_{0}^{1} 2 y d x
\]
is

$\frac{1}{3}$
$\frac{2}{3}$
$1$
$\frac{4}{3}$

GATE XE 2024 | Question: 6

Sun, 21 Jul 2024 16:42:40 +0000

For some integer $k$, the differential equation
\[
x^{2} \frac{d^{2} y}{d x^{2}}-3 x \frac{d y}{d x}+(k+2) y=0
\]
is transformed into $(D-2)^{2} y=0$, where $D=\frac{d}{d t}$ and $t=\log _{e} x$. Then, the value of $k$ is _________.

GATE XE 2022 | Question: 5

Fri, 17 Feb 2023 06:52:35 +0000

Let $y_{1}(x)$ and $y_{2}(x)$ be two linearly independent solutions of
\[
x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0, \quad x>0
\]
Let $W\left(y_{1}, y_{2}\right)(x)$ denote the Wronskian of $y_{1}(x)$ and $y_{2}(x)$ at $x$.
If $W\left(y_{1}, y_{2}\right)(1)=1$ then $W\left(y_{1}, y_{2}\right)(2)$ is equal to __________.