Question 456139
Divide both sides by ylny, to get

{{{lnx dx + dy/(ylny) = 0}}}.

<==> {{{lnx dx + d(lny)/lny = 0}}}

Integrate both sides:

{{{int(lnx, dx) + int(1/lny, d(lny)) = k}}} k an arbitrary constant.

After integration by parts (on the first integral), the equation becomes 

xlnx - x + ln(lny) = k.

Simplifying:

{{{ln(x^x) + ln(lny) = x+k}}}

<==> {{{ln(yx^x) = x+k}}}

<==> {{{yx^x = e^(x+k)}}}

<==> {{{yx^x = Ce^x}}} <==>  {{{y = C(e/x)^x}}}, where C is an arbitrary constant.