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Unformatted text preview: We solved the homogeneous heat equation with homogeneous Neumann boundary con-ditions in the practice problems for section 2.4. From this, guess a solution of the form u ( x, t ) = X n =0 A n ( t ) * eigenfunctions Proceed as in the previous problem to get an ODE of the form: dA n ( t ) dt + k n L 2 A n ( t ) = c n ( t ) Show that c n ( t ) = e-t if n = 0 e-2 t if n = 3 if n 6 = 0 , 3 Verify that with these three choices of c n ( t ), the solution of the ODE is: n 6 = 0 , 3 : A n ( t ) = A n (0) e-k ( n/L ) 2 t n = 0 : A ( t ) = A (0) + 1-e-t n = 3 : A 3 ( t ) = A 3 (0) e-k ( n/L ) 2 t + e-2 t-e-k ( n/L ) 2 t k (3 /L ) 2-2 1...
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- Summer '07