What is the solution of this differential equation?

[Esmaeil]

how can we solve this differential equation?
(ax+y)∂f/∂x + (ay+x)∂f/∂y =0 with this condition: if a=0 then f(x,y)=x^2 - y^2

 

[Chestermiller]

how can we solve this differential equation?
(ax+y)∂f/∂x + (ay+x)∂f/∂y =0 with this condition: if a=0 then f(x,y)=x^2 - y^2

$$df=\frac{∂f}{∂x}dx+\frac{∂f}{∂y}dy$$

At constant f, $$\frac{∂f}{∂x}dx+\frac{∂f}{∂y}dy=0$$

Or equivalently, $$\left(\frac{∂y}{∂x}\right)_f=-\frac{(∂f/∂x)}{(∂f/∂y)}$$

Chet

 

[JJacquelin]

The change of variables (in attachement) leads to a very simple PDE :