For $\displaystyle x$ $\displaystyle \in$ $\displaystyle R$ and $\displaystyle t > 0$ , define

$\displaystyle u(x, t) :=$ $\displaystyle \int_0^{x/\sqrt{t}} \! e^{-s^2} \, \mathrm{d}s$

Show that u satisfies the partial differential equation $\displaystyle 4{du/dt} = d^2u/dx^2$