If $y(x)$ is the solution of the differential equation $x d y-\left(y^2-4 y\right) d x=0 \text { for } x>0, y(1)=2,$ and the slope of the curve $y=y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is ____.
Solution and Explanation
The answer is 8.
Top Questions on Differential equations
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
- JEE Advanced - 2025
- Mathematics
- Differential equations
- For all $ x>0 $, let $ y_1(x), y_2(x), y_3(x) $ be the functions satisfying $$ \frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, $$ $$ \frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = \frac{1}{3}, $$ $$ \frac{dy_3}{dx} - \left(\frac{2 - x^3}{x^3}\right) y_3 = 0, \quad y_3(1) = \frac{3}{5e}, $$ respectively. Then $$ \lim_{x \to 0^+} \frac{y_1(x)y_2(x)y_3(x) + 2x}{e^{3x} \sin x} $$ is equal to __________.
- JEE Advanced - 2025
- Mathematics
- Differential equations
- Given the differential equation: \[ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad \text{with } y(1) = 0 \] Find the value of \(\frac{y^2(e)}{y(e^2)}\).
- JEE Advanced - 2025
- Mathematics
- Differential equations
- Find the solution $ \frac{d^2y}{dm^2} - k^3 \frac{dy}{dm} = y \cos m, \quad y(0) = 1 $
- MHT CET - 2025
- Mathematics
- Differential equations
- Let \( y = y(x) \) be the solution of the differential equation \[ \cos(x \log(\cos x))^2 \, dy + (\sin x - 3 \sin x \log(\cos x)) \, dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right) \] If \( y\left( \frac{\pi}{4} \right) = -1 \), then \( y\left( \frac{\pi}{6} \right) \) is equal to:
- JEE Main - 2025
- Mathematics
- Differential equations
Questions Asked in JEE Advanced exam
- Let a complex number \(z\) satisfy the equation: \[ |z - z_1| = 2 |z - z_2|,\quad \text{where } z_1 = 1 + 2i,\ z_2 = 3i \] Then choose the correct options:
- JEE Advanced - 2025
- complex numbers
A solid glass sphere of refractive index $ n = \sqrt{3} $ and radius $ R $ contains a spherical air cavity of radius $ \dfrac{R}{2} $, as shown in the figure. A very thin glass layer is present at the point $ O $ so that the air cavity (refractive index $ n = 1 $) remains inside the glass sphere. An unpolarized, unidirectional and monochromatic light source $ S $ emits a light ray from a point inside the glass sphere towards the periphery of the glass sphere. If the light is reflected from the point $ O $ and is fully polarized, then the angle of incidence at the inner surface of the glass sphere is $ \theta $. The value of $ \sin \theta $ is ____
- JEE Advanced - 2025
- Ray optics and optical instruments
The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity):
- JEE Advanced - 2025
- Oscillations
In a scattering experiment, a particle of mass $ 2m $ collides with another particle of mass $ m $, which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation $ \theta $ of the heavier particle, as shown in the figure, in radians is:
- JEE Advanced - 2025
- Elastic and inelastic collisions
A conducting square loop initially lies in the $ XZ $ plane with its lower edge hinged along the $ X $-axis. Only in the region $ y \geq 0 $, there is a time dependent magnetic field pointing along the $ Z $-direction, $ \vec{B}(t) = B_0 (\cos \omega t) \hat{k} $, where $ B_0 $ is a constant. The magnetic field is zero everywhere else. At time $ t = 0 $, the loop starts rotating with constant angular speed $ \omega $ about the $ X $ axis in the clockwise direction as viewed from the $ +X $ axis (as shown in the figure). Ignoring self-inductance of the loop and gravity, which of the following plots correctly represents the induced e.m.f. ($ V $) in the loop as a function of time:
- JEE Advanced - 2025
- Electromagnetic induction
Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations