In differential equation show that it is homogeneous and solve it. xdy/dx = y - xcos^2 ​(y/x)

Question

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

x\(\frac{dy}{dx}\) = y - xcos²\((\frac{y}{x})\)

Answer 1

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

= \((\frac{y}{x})-cos^2(\frac{y}{x})\)

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

⇒ 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