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
ANSWERED
asked 2021-03-09

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].

Answers (1)

2021-03-10

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.

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