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?