# Thread: Solving ODE by inspection

1. ## Solving ODE by inspection

Hello

I stopped at this one:

$y(x^2-y^2+1)dx-x(x^2-y^2-1)dy=0$

I opened the brackets:

$yx^2dx - y^3dx + ydx-x^3dy+xy^2dy+xdy=0$

It will be:

$yx^2dx - y^3dx-x^3dy+xy^2dy+d(xy)=0$

then ?!!!!

2. I multiplied the last equation by 3, to get:

$yd(x^3)-3y^3dx-3x^3dy+xd(y^3)+3d(xy)=0$

Now, I stopped.

3. Re-writing gives

$-(x^2-y^2)(xdy-ydx) + xdy+ydx = 0$

$-(x^2-y^2)x^2 d\left(\dfrac{y}{x}\right) + d(xy) = 0$

$(1-\frac{x^2}{y^2}) x^2y^2d\left(\dfrac{y}{x}\right) + d(xy) = 0$

$\left(1-\dfrac{1}{\left(\frac{y}{x}\right)^2}\right) d\left(\dfrac{y}{x}\right) + \dfrac{d(xy)}{(xy)^2} = 0$

4. Thanks ..

so the final answer will be :

$\dfrac{y}{x}+\dfrac{x}{y}-\dfrac{1}{xy}=0$

Right?

5. + c but that's what I got.