PDE $ u_{x}+u_{t}+f(x)*u=0$

Question

How would I solve this pde using characteristic line?
$u_{x}+u_{t}+f(x)u=0$---arbitrary function f
$u(x,0)=u_{0}(x)$---$u_{0}$ can be any value
$u(0,t)=\varphi(t)$---non-homogeneous
where
$u(x,t)\ge 0,\,\,\,\,0\le x\le l,\,\,\,\,t\ge 0$

Thank!

Answer 1

Suppose $x=x(t)$ is a characteristic line. Then $\frac{du}{dt} = u_x x' + u_t$. We conclude $x'(t) = 1$ -- the characteristic curves are $x=t+x_0$. However, $u$ is not constant along the characteristic -- it varies in a predictable way. In fact, $u'+f(t+x_0)u=0$ is an easily-solved ODE for $u(t)\equiv u(x(t),t)$, given that we know $f(x)$.