Question

ylny dx +(x-ln) dy=0

Answer 1

What are you solving for? @ketanthonylubguban
nice name btw =D

Answer 2

should that say ln(x) or ln(y)?

Answer 3

is it \[y(\ln y) dx +(x-\ln y) dy=0\]
or \[y(\ln y) dx +(x-\ln x) dy=0\] ??

Answer 4

if stgreen is right with \(y(\ln y)\,\mathrm dx +(x-\ln x)\,\mathrm dy=0\)

then the equation is variables separable
i.e. it can be re-arranged to be of this form;
\[\frac{\mathrm dy}{\mathrm dx}=f(y)g(x)\] (the derivative equals a function of one variable times the function of the other variable)
To solve from here,
separate the variables (/functions) like this
\[\frac{\mathrm dy}{f(y)}=g(x){\mathrm dx}\]

and integrate

\[\int\frac{\mathrm dy}{f(y)}=\int g(x){\mathrm dx}\]

don't forget to add the arbitrary constant of integration, (you only need one)

This constant will determine which of the solution curves you are on. If this was an initial value problem or boundary value problem you could determine the constant for that case. But otherwise one constant should remain in your solution as it is a first order DE.