In differential equation show that it is homogeneous and solve it. y^2 +(x^2 - xy)dy/dx = 0 ​

Question

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

y² +(x² - xy)\(\frac{dy}{dx}\) = 0

Answer 1

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

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

⇒ the given differential equation is a homogenous equation. 

The solution of the given differential equation is : 

Put x = vy

Integrating both the sides we get:

⇒ y = x(ln|y| + c) 

Ans: y = x(ln|y| + c)