Consider a heat equation for the temperature u of a rod of length 1:(adsbygoogle = window.adsbygoogle || []).push({});

u_{t}= u_{xx}, 0 < x < 1, t > 0 with boundary conditions u_{x}(0,t) = 0 & u(1,t) = 0. I derived X_{n}(x) = cos((n+1/2)[tex]\pi[/tex]x) using separation of variables.

How do I show that [tex]\int_{0}^1[/tex] X_{n}(x)X_{m}(x) dx = 1/2 if m = n and 0 if m [tex]\neq[/tex] n.

I used the product to sum formula: cos(A)cos(B) = cos(A+B)/2 + cos(A-B)/2 to get 1/2cos((n+m+1)[tex]\pi[/tex]x) 1/2cos((n-m)[tex]\pi[/tex]x) but I am stuck after that. Someone help, am I even on the right track.

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# Heat equation

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