I have a PDE which takes the form:
$$ \frac{\partial u}{\partial t} +c\frac{\partial u}{\partial x} = f(x,t) $$
So I understand how to get to the point where we make a coordinate change, $\xi = x-ct$ and $t = \tau$ to give $u(x,t) = U(\xi,\tau)$, and so the above PDE becomes:
$$ \frac{\partial U}{\partial \tau} = F(\xi,\tau) $$
But then where my understanding breaks down is when explicitly integrating gives:
$$ U(\xi,\tau) = \int F(\xi,\tau)d\tau + G(\xi) $$
Where $G(\xi)$ is the "constant" of integration. Why is this "constant" a function of $\xi$? Surely this constant can also be just a normal constant. Is this just a way of hedging my bets and saying, yeah sure it could be just a constant, but it ALSO could be a function of $\xi$ since we only integrated with respect to $\tau$ and if I differentiated this thing again with respect to $\tau$ it's going to disappear anyway?