In my differential equations class I have been asked to find and sketch the characteristics for the equation:

partial du/dy + u * partial du/dx = 0 , for all x and y>=0

ad subject to the intial data u(x,y=0) = 2, x<0 and 1, x>0

So far I have managed to find the characteristic equations but one of them is giving me some trouble

I have that dx/dt= u, I know that this has to be intergrated by seperation
i.e. int (dx/u) = int (dt) but I am unsure how to do this since u depends on x but Im not sure how

Any help would be greatly appreciated,