Say we want to solve this on the interval x>0.

And define a new function u=y/x

Meaning, y=ux

Differenciate both sides (use the product rule!)

y'=u+u'x

Substitue that into the differencial equation:

u+u'x = u - exp(u)

Thus,

u'x=-exp(u)

Thus,

u'*exp(-u) = -1/x

Integrate both sides,

INT u'*exp(-u) dx = INT -1/x dx

Thus, (substitution rule)

-exp(-u) = -ln x +C

Multiply by (-1),

exp(-u) = ln x +C

Then,

-u = ln ( ln x +C) for C>=0

Thus,

-y/x = ln (ln x +C)

Thus,

y= - x * ln (ln x +C) for C>=0