Question

Determine an appropriate solution to solve xdy/dx=yln(xy)

Answer 1

`x(\mathrm dy)/(\mathrm dx)=y\ln(xy)`

Let
`v=xy`, so that
`(\mathrm dv)/(\mathrm dx)=y+x(\mathrm dy)/(\mathrm dx)`, or
`(\mathrm dv)/(\mathrm dx)-\frac vx=x(\mathrm dy)/(\mathrm dx)`. Then you have


`(\mathrm dv)/(\mathrm dx)-\frac vx=\frac vx\ln v`

`(\mathrm dv)/(v(\ln v+1))=\frac{\mathrm dx}x`

Integrate both sides to get


`\ln|\ln v+1|=\ln|x|+C`

`\ln v+1=Cx`

`v=e^(Cx-1)`

then back-substitute to find the solution for
`y`.


`xy=e^(Cx-1)\implies y=\frac{e^(Cx-1)}x`