Solve `x(dy)/(dx)=y(logy-logx+1)`

Options

A. `log((x)/(y))=Cy`
B. `log((y)/(x))=Cx`
C. `x log((y)/(x))=Cy`
D. `ylog((x)/(y))=Cx`

Answer

B

Explanation

We have,
`x(dy)/(dx)=ylog((y)/(x))+y`
Putting `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)`, we get
`v+x(dv)/(dx)=vlog v +v`
`rArr" "(1)/(v log v) dv=(1)/(x)dx`
`rArr" "log(logv)=logx+logC" [On integrating]"`
`rArr" "logv=Cx`
`rArr" "log((y)/(x))=Cx`, which is the required solution.