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

Question

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

x²\(\frac{dy}{dx}\) = x² + xy + y²

Answer 1

⇒ x²\(\frac{dy}{dx}\) = x² + xy + y²

⇒ the given differential equation is a homogenous equation. 

The solution of the given differential equation is : 

Put y = vx

⇒ \(\cfrac{dv}{1+(v)^2}\) = \(\cfrac{dx}x\)

Integrating both the sides we get:

⇒ \(\int\cfrac{dv}{1+(v)^2}\) = \(\int\cfrac{dx}x\) + c

⇒ tan ^- v = ln|x| + c 

Resubstituting the value of y = vx we get 

⇒ tan ^- (y/x) = ln|x| + c 

Ans: tan ^- (y/x) = ln|x| + c