# How to solve $(1+xy) y\,dx + (1-xy)x\,dy = 0$ by sepatation, as a linear equation or homogeneous?

How to solve this ODE?

$$(1+xy) y\,dx + (1-xy)x\,dy = 0$$

We have been taught only these 3 types of problems: separable, linear equations or homogeneous. I tried using these methods, but I don't think this is a homogeneous or a linear differential equation. I tried separating the variables but couldn't. I strongly believe that this can be only solved by separating variables otherwise the question is wrong.

• Formatting tips here. – Em. Feb 21 '16 at 6:10

## 1 Answer

HINT :

A part $y(x)=0$ and $x(y)=0$ which both are trivial solutions : $$y'(x)= -\frac{(1+xy)y}{(1-xy)x}$$ $t(x)=xy \quad \implies \quad t'=y+xy'=\frac{t}{x}-\frac{(1+t)t}{(1-t)x} = -\frac{2t}{(1-t)x}$

$$dt = -\frac{2t}{(1-t)x}dx$$

This is a separable ODE : I suppose that you know how to continue.

• Thank you, I don't know why i was not able to do it, I tried that too, something must have went wrong – rishabh gupta Feb 21 '16 at 6:27
• Which will again lead the marvelous Lambert function ! – Claude Leibovici Feb 21 '16 at 7:03
• Without the miraculous Lambert function, one have to be satisfied with an implicit form of solution ! – JJacquelin Feb 21 '16 at 11:12