What to do with the $y$ term when solving $yu_{y} + uu_{x} =u-y$ I have an initial value problem that looks like this
$$yu_{y} + uu_{x} =u-y; \qquad u(x,1)=x$$
I think I can just use the method of characteristics to solve it, which I would be perfectly capable of doing, except I am not sure how to treat the $-y$ 
Perhaps I am missing something obvious, but it has me stuck.

The equation of the characteristics is
$$
\frac{dy}{y}=\frac{dx}{u}=\frac{du}{u-y}.
$$
