In differential equation show that it is homogeneous and solve it. dy/dx = x^2 + y^2/2xy ​

Question

In differential equation show that it is homogeneous and solve it.

\(\frac{dy}{dx}\) = \(\frac{x^2+y^2}{2xy}\)

dy/dx = x² + y²/2xy

Answer 1

\(\frac{dy}{dx}\) = - \(\frac{x^2+y^2}{2xy}\)

⇒ \(\frac{dy}{dx}\) = - \((\frac{y}{2x})^{-1}\) - \((\frac{y}{2x})\)

⇒ \(\frac{dy}{dx}\) = \(f(\frac{y}{x})\)

⇒ the given differential equation is a homogenous equation. 

The solution of the given differential equation is : 

Put y = vx

Integrating both the sides we get:

Resubstituting the value of y = vx we get

⇒ In\(|(\frac{y}{x})^2-4|\) = In|x| + In|c|

⇒ (x² - y²) = cx 

Ans: (x² - y²) = cx