Let the solution curve y = y(x) of the differential equation (4 + x2)dy – 2x(x2 + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _______.
Let the solution curve y = y(x) of the differential equation (4 + x2)dy – 2x(x2 + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _______.
Correct Answer: 12
Solution and Explanation
The correct answer is 12
(4 + x2) dy – 2x(x2 + 3y + 4)dx = 0
\(⇒\frac{dy}{dx}=(\frac{6x}{x^2+4})y+2x\)
\(⇒\frac{dy}{dx}−(\frac{6x}{x^2+4})y=2x\)
I.F. \(= e^{−3 In(x^2+4)}=\frac{1}{(x^2+4)^3}\)
\(So, \frac{y}{(x^2+4)^3}=∫\frac{2x}{(x^2+4)^3}dx+c\)
\(⇒y=−\frac{1}{2}(x^2+4)+c(x^2+4)^3\)
When x = 0, y = 0 gives
\(c=\frac{1}{32}\)
So, for x = 2,y = 12
Top Questions on Differential equations
- Let \( y = y(x) \) be the solution of the differential equation \( \frac{dy}{dx} + 3(\tan^2 x) y + 3y = \sec^2 x \), with \( y(0) = \frac{1}{3} + e^3 \). Then \( y\left(\frac{\pi}{4}\right) \) is equal to
- JEE Main - 2025
- Mathematics
- Differential equations
- Let g be a differentiable function such that $ \int_0^x g(t) dt = x - \int_0^x tg(t) dt $, $ x \ge 0 $ and let $ y = y(x) $ satisfy the differential equation $ \frac{dy}{dx} - y \tan x = 2(x+1) \sec x g(x) $, $ x \in \left[ 0, \frac{\pi}{2} \right) $. If $ y(0) = 0 $, then $ y\left( \frac{\pi}{3} \right) $ is equal to
- JEE Main - 2025
- Mathematics
- Differential equations
- If a curve $ y = y(x) $ passes through the point $ \left(1, \frac{\pi}{2}\right) $ and satisfies the differential equation $$ (7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5, \quad x \geq 1, \text{ then at } x = 2, \text{ the value of } \cos y \text{ is:} $$
- JEE Main - 2025
- Mathematics
- Differential equations
- General solution of the differential equation \[ \frac{dy}{dx} + y \tan x = \sec x \quad \text{is:} \]
- KCET - 2025
- Mathematics
- Differential equations
- The general solution of the differential equation \( \frac{dy}{dx} = 3x^2 \) is:
- MHT CET - 2025
- Mathematics
- Differential equations
Questions Asked in JEE Main exam
- A force of 49 N acts tangentially at the highest point of a sphere (solid of mass 20 kg) kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is:
- JEE Main - 2025
- Quantum Mechanics
Match List-I with List-II: List-I
- JEE Main - 2025
- Classification Of Organic Compounds
- Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length a. When released, they interact only through gravitational force and collide after a time T = 4 seconds. If the sides of the triangle are increased to length 2a and also the masses of the spheres are made 2m, then they will collide after ______ seconds.
- JEE Main - 2025
- Gravitation
- A photo-emissive substance is illuminated with a radiation of wavelength $ \lambda_i $ so that it releases electrons with de-Broglie wavelength $ \lambda_e $. The longest wavelength of radiation that can emit photoelectron is $ \lambda_0 $. Expression for de-Broglie wavelength is given by : (m : mass of the electron, h : Planck's constant and c : speed of light)
- JEE Main - 2025
- Photoelectric Effect
The dimension of $ \sqrt{\frac{\mu_0}{\epsilon_0}} $ is equal to that of: (Where $ \mu_0 $ is the vacuum permeability and $ \epsilon_0 $ is the vacuum permittivity)
- JEE Main - 2025
- Electromagnetic waves
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