Method of Characteristics Q

Please see attached.

So after we have inverted x and y in terms of s and t, we have following, x=e^s and y=(e^s)(s+t) so:

s = lnx
t = (y/x) - lnx

Then along the (t)th characteristic curve, du/ds = 1 => u = s + k

This is where the problem arises. We know u(1,y) = y. So do we put that into the above to find k? So i.e. have u = lnx + k. So at x=1, u=y so y=k => u = lnx + y.

OR do we use u(1,y) = y in terms of t and then substitute to find k? If so how do we do it?

Thanks.