First order method of characteristics

I am tasked with solving

$$cos(y)frac{partial u}{partial x} + x frac{partial u}{partial y} = frac{x}{u}$$

And can’t seem to get past the characteristics, namely:

$$frac{dx}{cos(y)} = frac{dy}{x}$$

and

$$frac{du}{x/u} = frac{dx}{cos(y)}.$$

How might I proceed here?