Sub in the point (1, 1):
(1/4) ln|1 - 2| - (1/4) ln|1 + 2| = ln|1| + C
(1/4) ln(1) - (1/4) ln(3) = ln(1) + C
(-1/4)ln(3) = C
ln( 3^(-1/4) ) = C
(1/4) ln|y - 2| - (1/4) ln|y + 2| = ln|x| + ln( 3^(-1/4) )
(1/4) [ln|y - 2| - ln|y + 2|] = ln|3^(-1/4) x|
(1/4) [ln|(y - 2) / (y + 2)|] = ln|3^(-1/4) x|
ln|{ (y - 2) / (y + 2) }^(1/4)| = ln|3^(-1/4) x|
[ | (y - 2) / (y + 2) | ]^(1/4) = | 3^(-1/4) x |
| (y - 2) / (y + 2) | = (1/3)x^4
Now there are a couple ways to isolate y because of the absolute value brackets, either:
y = (6 - 2x^4) / (x^4 + 3)
y = (6 + 2x^4) / (x^4 - 3)
But we need (1, 1) to be a point, which means that we must have the first one:
y = (6 - 2x^4) / (x^4 + 3)
Done!