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)