Question

Solve: $y(x-1)dx+(y-1)xdy=0$.

Answer 1

Given,

$y(x-1)dx+(y-1)dy=0$

$\frac{x-1}{x}dx=-\frac{y-1}{y}dy$

integrating on both sides, we get,

$\int \frac{x-1}{x}dx=-\int \frac{y-1}{y}dy$

$x-\log x=-y+\log y+c$

$x+y=\log x+\log y+c$

$x+y=\log xy+c$