Question

Solve dy=(y²-1)dx. Ans: ln((y-1)/(y+1))=2x+c

Answer 1

Divide both sides by $y^2-1$ and integrate:

$\dfrac{\mathrm dy}{y^2-1}=\mathrm dx$

To integrate left side, first expand into partial fraction:

$\dfrac1{y^2-1}=\dfrac12\left(\dfrac1{y-1}-\dfrac1{y+1}\right)$

$\displaystyle\int\frac{\mathrm dy}{y^2-1}=\int\mathrm dx$

$\dfrac12\left(\ln|y-1|-\ln|y+1|\right)=x+C$

$\ln\left|\dfrac{y-1}{y+1}\right|=2x+C$