I have to solve the following quasilinear equation using the method of characteristics:

$\displaystyle x u_x + y u_y +yu=xy$

$\displaystyle u|_{y=x^2}=sin x$

$\displaystyle \frac{dx}{dt}=x$ gives me $\displaystyle ln x=t + c_1$

$\displaystyle \frac{dy}{dt}=y$ gives me $\displaystyle ln y=t+c_2$

But what do I do with $\displaystyle \frac{du}{dt}=xy - yu$?

Thank you.