I am always very dubious when people beg for full solutions after I read things like "I'm stuck halfway" or "I've gone as far as I can" without even showing what they have done to prove they've put in some effort. When I see that, I'll put in some effort to solve this DE...
It seems that you have no nowledge of integration .
How you took out of the integral the variables y and x
First you did correct to separate the variables after all this D.E can be done this way.
now try to integrate the fractions (y/y^2+1)dy and (x/x^2+1)dx both are the same....
Manipulate the differential dx and make it d(x^2+1) and divide the integral by 1/2 the integral will look like this
(1/2)integral(d(x^2+1)/(x^2+1)) which of course give you (1/2)Ln(y^2+1) = (1/2)ln(x^2+1)+k or you may put k=(1/2)lnc
so that y^2+1=c(x^2+1)
Use the conditioons given to calculate the constant c which is 1/2
therefore the solution of the D.E IS y^2=1/2(x^2+1)-1
check the calculations for any typing error .