Question

Find Integrating Fector:
ylogy dx + ( x-logy )dy=0

Answer 1

Rewrite the ODE as


`y\log y\,\mathrm dx+(x-\log y)\,\mathrm dy=0\iff y\log y(\mathrm dx)/(\mathrm dy)+x=\log y`

`(\mathrm dx)/(\mathrm dy)=\frac1{y\log y}x=\frac1y`

so that it is now linear in
`x`. An integrating factor would


`\mu(y)=\exp\left(\displaystyle\int(\mathrm dy)/(y\log y)\right)=e^(\log(\log y))=\log y`

Multiply both sides by
`\mu(y)` to get


`\log y(\mathrm dx)/(\mathrm dy)=\frac1yx=\frac{\log y}y`

`(\mathrm d)/(\mathrm dy)[x\log y]=\frac{\log y}y`

`x\log y=\displaystyle\int\frac{\log y}y\,\mathrm dy`

`x\log y=\frac12\log^2y+C`

`x=\frac12\log y+\frac C{\log y}`