Chapter 1 - Introduction - 1.3 Classification of Differential Equations - Problems - Page 25: 28

$u_{x}=\dfrac{-2x}{4\alpha^2t}(\dfrac{\pi}{t})^{1/2}e^{\dfrac{-x^2}{4\alpha^2t}}$ $u_{xx}=\dfrac{-2(4\alpha^2t-2x^2)}{(4\alpha^2t)^2}(\dfrac{\pi}{t})^{1/2}e^{\dfrac{-x^2}{4\alpha^2t}}$ $u_{xx}=\dfrac{-1}{2\alpha^2}(\sqrt\pi(t-2\frac{x^2}{4\alpha^2}))t^{-5/2}e^{\dfrac{-x^2}{4\alpha^2t}}$ $u_t=\dfrac{-1}{2}(\sqrt\pi(t-2\frac{x^2}{4\alpha^2}))t^{-5/2}e^{\dfrac{-x^2}{4\alpha^2t}}$ $\alpha^2u_{xx}=u_t$