Help Solving for 'y' when 'y' is in numerator and denominator

Question

I'm taking a differential equations college course, and I'm embarrassed to say that I'm a bit hung up on some algebra with a particular problem. I'm having issues getting an implicit solution into explicit form by hand.

$\frac{y-2}{y+2}=\pm ce^{4x}$

My algebra skills are extremely rusty, and I'm banging my head against the wall trying to figure out how to solve this equation explicitly for $y$. I can get a solution on my calculator, but I would prefer to understand how to come to the solution by hand. Any help would be appreciated.

Answer 1

let $$\pm ce^{4x}=r$$ then we have $$\frac{y-2}{y+2}=r$$ multiplying by $y+2$ we get $$y-2=r(y+2)$$ $$y-2=ry+2r$$ $$y-ry=2+2r$$ $$y(1-r)=2(1+r)$$ $$y=\frac{2(1+r)}{1-r}$$ back substituion of $r$ gives the searched result

Answer 2

$$\frac{y-2}{y+2}=\pm ce^{4x}\\ \frac{y+2-4}{y+2}=\pm ce^{4x}\\ 1-\frac{4}{y+2}=\pm ce^{4x}\\ 1-\pm ce^{4x}=\frac4{y+2}\\ \frac1{1-\pm ce^{4x}}=\frac{y+2}4\\ \frac4{1-\pm ce^{4x}}=y+2\\ \frac4{1 \mp ce^{4x}}-2=y\\ y = \frac{2\pm2ce^{4x}}{1\mp ce^{4x}}$$