Question:
When $ y = vx $ , $ y $ and $ x $ are variables, the differential equation $ \frac{dy}{dx}=\frac{2xy}{x^{2}-y^{2}} $ reduces to
When $ y = vx $ , $ y $ and $ x $ are variables, the differential equation $ \frac{dy}{dx}=\frac{2xy}{x^{2}-y^{2}} $ reduces to
Updated On: Jun 23, 2024
- $ \frac{1-v^{2}}{v+v^{3}}dv=\frac{2}{x}dx $
- $ \frac{1-v^{2}}{v+v^{3}}dv=\frac{1}{x}dx $
- $ \frac{1+v^{2}}{v+v^{3}}dv=\frac{1}{x}dx $
- $ \frac{1-v^{3}}{v+v^{3}}dv=\frac{1}{x}dx $
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The Correct Option is B
Solution and Explanation
We have, $\frac{dy}{dx} = \frac{2xy}{x^2 - y^2}\,\,\,...(i)$
Put $ y = vx $
$\Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx}$
$\therefore $ Equation $(i)$ becomes,
$ v + x \frac{dv}{dx} = \frac{2x^2v}{x^2 - v^2 x^2}$
$\Rightarrow x \frac{dv}{dx} = \frac{v + v^3}{1-v^2}$
$\Rightarrow \left(\frac{1-v^{2}}{v + v^{3}}\right) dv = \frac{dx}{x}$
Put $ y = vx $
$\Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx}$
$\therefore $ Equation $(i)$ becomes,
$ v + x \frac{dv}{dx} = \frac{2x^2v}{x^2 - v^2 x^2}$
$\Rightarrow x \frac{dv}{dx} = \frac{v + v^3}{1-v^2}$
$\Rightarrow \left(\frac{1-v^{2}}{v + v^{3}}\right) dv = \frac{dx}{x}$
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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