Practice exam solutions 3

You may assume that when you set the equations equal

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Unformatted text preview: assume that when you set the equations equal to a separation constant − λ , the cases λ = 0 and λ = −α 2 give trivial solutions. This is known as the Heat-Neumann model. Assume , = = , ,0 = , = 0, and plug this into PDE to get = Case 1: λ = 0 → = 0 Trivial solution Case 2: λ = −α 2 → = 0 Trivial solution Case 3: : λ = α 2 =−λ =0 4 5 5.) [10 points] A string of length π has tension constant a and has its two ends clamped at the x-axis. Assuming the string is released from rest, the vertical displacement u(x,t) of the string is modeled by = 0, , Suppose the string has initial shape = , = 0, ,0 = 0 , 0 = sin for some integer...
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