Question

Show by substitution that u(x,t)=cos(απx)e^−α^2π^2t is a solution of the heat equation ut=uxx on any interval [0, L].

Integrals

Show by substitution that $$u(x,t)=\cos(απx)e^−α^2π^2t$$ is a solution of the heat equation $$ut=uxx$$ on any interval [0, L].

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