1

just realized I had a typo (missing a parenthesis):

v = y/x

=> y=vx => y' = v + v'x

xy' = y - xe^(y/x) => y' = y/x - e^(y/x)

=> v + v'x = v - e^v

=> v'x = -e^v

=> (-e^-v)dv = dx/x

=> e^(-v) = ln (x) + c

=> ln(e^-v) = ln(ln (x) +c)

=> v = -ln(ln(x) +c)

=> y = -xln(ln(x)+c)

y(1)=-2 => c = e^-2

=> y = -xln(ln(x) + e^-2)

v = y/x

=> y=vx => y' = v + v'x

xy' = y - xe^(y/x) => y' = y/x - e^(y/x)

=> v + v'x = v - e^v

=> v'x = -e^v

=> (-e^-v)dv = dx/x

=> e^(-v) = ln (x) + c

=> ln(e^-v) = ln(ln (x) +c)

=> v = -ln(ln(x) +c)

=> y = -xln(ln(x)+c)

y(1)=-2 => c = e^-2

=> y = -xln(ln(x) + e^-2)