Help solving an ODE with Separation of Variables.

So I did this with bernouli substitution on an exam and got partial credit because the question said to do this with separation of variables. I feel like this shouldn't be that hard, but I cant for the life of me figure it out. The ODE is: $$y'=y+y ^{-1}$$

Question

Help solving an ODE with Separation of Variables.

So I did this with bernouli substitution on an exam and got partial credit because the question said to do this with separation of variables. I feel like this shouldn't be that hard, but I cant for the life of me figure it out. The ODE is: $$y'=y+y ^{-1}$$

anonymous · Answer

I started with $$\frac{ dy }{ dt } -y = \frac{ 1 }{ y }$$

anonymous · Answer

Is  there some sort of algebra trick I am not seeing?  I can't get y by itself to do separation of variables..

ikram002p · Answer

sound serable to me

anonymous · Answer

oh man, i think I get it.

ikram002p · Answer

yeah :) let y and dy in one side 
and dt in the other side , so what u got ?

anonymous · Answer

$$(y+y^-1 ) dt$$ ug i feel dumb

anonymous · Answer

$$\frac{ dy }{ dx }=y+y ^{-1}=y+\frac{ 1 }{ y }=\frac{ y^2+1 }{ y }$$
separating the variables
$$\frac{ y }{ y^2+1 }dy=dx$$
integrating

anonymous · Answer

Ya, ug, thanks a lot, that wasn't bad at all.... ugggggg!

anonymous · Answer

Thanks for looking guys

anonymous · Answer

$$\int\limits \frac{ 2y dy }{ y^2+1 }=\int\limits 2 x dx+c$$
can you solve further?

anonymous · Answer

$$\ln \left( y^2+1 \right)=2 \frac{ x^2 }{ 2 }+c$$