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:

`(\mathrm dy)/(y^2-1)=\mathrm dx`

To integrate left side, first expand into partial fraction:

`\frac1{y^2-1}=\frac12\left(\frac1{y-1}-\frac1{y+1}\right)`

`\displaystyle\int(\mathrm dy)/(y^2-1)=\int\mathrm dx`

`\frac12\left(\ln|y-1|-\ln|y+1|\right)=x+C`

`\ln\left|(y-1)/(y+1)\right|=2x+C`