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Question Number 168977 by MikeH last updated on 23/Apr/22

check that the function  u(x,t) = exp{−((n^2 α^2 π^2 )/L^2 )t} sin((nπx)/L)  n = 1,2,... satisfy the heat equation  heat equation  α^2 (∂^2 u/∂x^2 ) = (∂u/∂t), 0 < x < L

$$\mathrm{check}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function} \\ $$ $${u}\left({x},{t}\right)\:=\:\mathrm{exp}\left\{−\frac{{n}^{\mathrm{2}} \alpha^{\mathrm{2}} \pi^{\mathrm{2}} }{{L}^{\mathrm{2}} }{t}\right\}\:\mathrm{sin}\frac{{n}\pi{x}}{{L}} \\ $$ $${n}\:=\:\mathrm{1},\mathrm{2},...\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{heat}\:\mathrm{equation} \\ $$ $$\boldsymbol{\mathrm{heat}}\:\boldsymbol{\mathrm{equation}} \\ $$ $$\alpha^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }\:=\:\frac{\partial{u}}{\partial{t}},\:\mathrm{0}\:<\:{x}\:<\:{L} \\ $$

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