Chapter 2.2 problem 2 equation (9): $u_t + yu_x + xu_y = x$

and we get $\frac{dt}{1} = \frac{dx}{y} = \frac{dy}{x} = \frac{du}{x}$

By solving $\frac{dx}{y} = \frac{dy}{x}$ first we get $C = x^2 - y^2$

But we solving for either $\frac{dt}{1} = \frac{dx}{y}$ or $\frac{dt}{1} = \frac{dy}{x}$ we need to substitute $x$ for $y$ or the other direction, so in this case we will have a square root function like $x = \pm \sqrt{y^2 + C}$

I am not entirely sure whether or not we should proceed with the $\pm$ sign here or is there another easier approach to this question?