Question:
If $x \frac{dy}{dx} =y \left(\log \,y - \log \,x\right)$ then the solution of the equation is
If $x \frac{dy}{dx} =y \left(\log \,y - \log \,x\right)$ then the solution of the equation is
Updated On: Jul 6, 2022
- $\log (\frac {x} {y}) =cy $
- $\log (\frac {y} {x}) =cx $
- $x \log (\frac {y} {x}) =cy $
- $y \log (\frac {x} {y}) =cx $
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
$x \frac{dy}{dx} =y \left(\log \,y - \log \,x+1\right)$
$\Rightarrow \frac{dy}{dx} = \frac{y}{x}\left(log \frac{y}{x}+1\right)$.
Put $\frac{y}{x} = z$
$\therefore \frac{dy}{dx}=x\frac{dz}{dx}+z$
$\therefore z+x \frac{dz}{dx}= z\left(log\,z+1\right)$
$\Rightarrow x \frac{dz}{dx} \,z\,log\,z$
$\Rightarrow \frac{dz}{z\,log\,z} = \frac{dy}{x}$
$\Rightarrow log \,\left(log \,z\right) = log \,x + log\, C$
$\Rightarrow log \,z = Cx$
$\Rightarrow log \left(\frac{y}{x}\right) = Cx$
Was this answer helpful?
0
0
Top Questions on Differential equations
- General solution of the differential equation \[ \frac{dy}{dx} + y \tan x = \sec x \quad \text{is:} \]
- KCET - 2025
- Mathematics
- Differential equations
- If 'a' and 'b' are the order and degree respectively of the differentiable equation \[ \frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + x^4 = 0, \quad \text{then} \, a - b = \, \_ \_ \]
- KCET - 2025
- Mathematics
- Differential equations
- Let \( y = f(x) \) be the solution of the differential equation\[\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, \quad -1 < x < 1\] such that \( f(0) = 0 \). If \[6 \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha\] then \( \alpha^2 \) is equal to __________.
- JEE Main - 2025
- Mathematics
- Differential equations
- If \( x = f(y) \) is the solution of the differential equation \[ (1 + y^2) + (x - 2e^{\tan^{-1}y}) \frac{dy}{dx} = 0, \quad y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right), \] with \( f(0) = 1 \), then \( f\left( \frac{1}{\sqrt{3}} \right) \) is equal to:
- JEE Main - 2025
- Mathematics
- Differential equations
- Let \( x = x(y) \) be the solution of the differential equation \( y^2 dx + \left( x - \frac{1}{y} \right) dy = 0 \). If \( x(1) = 1 \), then \( x\left( \frac{1}{2} \right) \) is :
- JEE Main - 2025
- Mathematics
- Differential equations
View More Questions
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