We will find ut and uxx, \(ut=(d/dt)\cos(\alpha \pi x)e^{a2}\pi^{2t}=-a^{2}\pi^{2}\cos(\alpha \pi x)e^{a2}\pi^{2t}\)

\(ux=(d/dt)\cos(\alpha \pi x)e^{a2} \pi^{2t}=-\alpha \pi \sin(\alpha \pi x)e^{a2}\pi^{2t}\)

\(uxx=(d/dx)(-\alpha \pi \sin(\alpha \pi x)e^{a2} \pi^{2t})=-a^{2} \pi^{2} \cos( \alpha \pi x)e^{a2} \pi^{2t}\)

Therefore, ut=uxx, so the statement is proved.