With the solution you have we find that
and dividing gives
or
.
From your solution .
Thus,
.
Imposing the initial curves gives your answer.
Asked to derive the general solution in the given form which I have done and is as attached...
The intial data curve is
For clarity the last line of attachment is as above which I know is correct...
How do I show that for the special case u=x when y=1?
I believe this requires getting u in terms of x and y but I dont know how to do that with variables in the above equations...