Question

(x-y)dx+xdy=0

Answer 1

\[\left( x-y \right)dx+xdy=0,\frac{ dy }{dx }=\frac{ y-x }{ x }\]
put y=vx
\[\frac{ dy }{dx }=v+x \frac{ dv }{ dx }\]
\[v+x \frac{ dv }{dx }=\frac{ vx-x }{ x }=v-1,x \frac{ dv }{ dx }=v-1-v=-1\]
\[dv=-\frac{ dx }{ x }\]
integrate both sides and then replace v by y/x