It's simply the product rule. You are differentiating with respect to s. Presumably, x and y are functions of s but u, here, is independent of s so we can treat it as a constant. The derivative of is .
Sorry, I should have mentioned that all 3 functions, x, y and u are functions of s here.
I understand the product rule, but I don't have a clue where the (x+u)/y sprung from.
My understanding of this problem is that we're trying to find two functions
= constant
= constant
as the solution of the equation satisfies = 0
Since = 0,
(x+u)/y is a constant, and so there's .
I just don't understand how we got there.
In the notes I have, there is alternative way of solving these quasilinear equations, which involves integrating the characteristic equations, then parametrising the initial line and setting s = 0. Does this sound familiar to anyone?