I'm trying to show that the function $$u(x,t) = \int^t_0 s(x + b(\tau - t), \tau) d\tau$$ satisfies the partial differential equation $$u_t + bu_x = s(x,t).$$
I start by finding $$u_t(x,t) = \frac{\partial}{\partial t}\int^t_0 s(x + b(\tau - t), \tau) \, d\tau =s(x,t)$$ and then $$u_x(x,t) = \frac{\partial}{\partial x}\int^t_0 s(x + b(\tau - t), \tau) \, d\tau$$ $$= \int^t_0 \frac{\partial}{\partial x}s(x + b(\tau - t),\tau) \, d\tau$$ and this is where I get stuck.
Am I on the right track?