I am trying to solve following equation:

$$u_t + u_x + \frac{u}{x} = 0$$

With initial condition: $$u(x,0) = 0$$

And with boundary condition given at x = 15: $$u(15,t) = sin (wt)$$

I tried to transform it to the form of: $$u_t + u_x =- \frac{u}{x}$$

And solve it with method of characteristics. However, 1) I am not sure how to treat x on the right side (all online examples have source term either x or u, never both). 2) I am not sure when and how to insert this boundary condition in this equation?

My kind of solution always include e, and solution provided by tutor does not contain any e terms.

I tried following this example Advection Equation with $f(x)\cdot u(x,t)$ source term, however I did not understand it completely, specifically part "which can easily be integrated to yield..."

$$dt = dx = \frac{du}{-u/x}.$$
From 1st and 2nd we have $t-x = c_1$ is a characteristic of the PDE. From 2nd and 3rd we would have that $u = c_2/x$ is the other characteristic curve. Put $c_2$ as a function of $c_1$ and you are done!