Divide both sides by [tex]y^2-1[/tex] and integrate:
[tex]\dfrac{\mathrm dy}{y^2-1}=\mathrm dx[/tex]
To integrate left side, first expand into partial fraction:
[tex]\dfrac1{y^2-1}=\dfrac12\left(\dfrac1{y-1}-\dfrac1{y+1}\right)[/tex]
[tex]\displaystyle\int\frac{\mathrm dy}{y^2-1}=\int\mathrm dx[/tex]
[tex]\dfrac12\left(\ln|y-1|-\ln|y+1|\right)=x+C[/tex]
[tex]\ln\left|\dfrac{y-1}{y+1}\right|=2x+C[/tex]
