Be sure to give the intermediate steps a separate

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). Be sure to give the intermediate steps: (a) separate variables, (b) write down problems and solve for X ( x ) (using information from the Given section), (c) solve for T n ( t ), (d) put things together, impose the IC, (e) use orthogonality of cos ( ( n - 1 2 ) πx ) (see Given section) to find A n in terms of f ( x ). Substitute for f ( x ) from Part 2. You may use (without proof) the following integrals, for any integer n , 1 0 (1 - x 4 ) cos ( ( n - 1 2 ) πx ) dx = ( - 1) n +1 96( π 2 (2 n - 1) 2 - 8) (2 n - 1) 5 π 5 , 1 0 cos ( ( n - 1 2 ) πx ) dx = 2( - 1) n +1 (2 n - 1) π Page 3 of 5 Please go to the next page. . . [0.5 points] [0.5 points] [0.5 points] [0.5 points] [0.5 points] [0.5 points]
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Solutions to Test 1 Solutoin: Using separation of variables, we let v ( x, t ) = X ( x ) T ( t ) and substitute this into the PDE to obtain X X = T T = - λ where λ is a constant because the left hand side depends only on x and the middle only depends on t . The Sturm-Liouville problem for X ( x ) is X + λX = 0; X (0) = 0 = X (1) whose solution is (given), X n ( x ) = cos ( ( n - 1 2 ) πx ) , λ n = ( n - 1 2 ) 2 π 2 , n = 1 , 2 , . . . The equations for T ( t ) are T n ( t ) = A n e - ( n - 1 2 ) 2 π 2 t and this gives the solution v n ( x, t ) to the PDE v n ( x, t ) = X n ( x ) T n ( t ) = A n cos ( ( n - 1 2 ) πx ) e - ( n - 1 2 ) 2 π 2 t for constants A n . Summing all v n ( x, t ) together gives v ( x, t ) = n =1 v n ( x, t ) = n =1 A n cos ( ( n - 1 2 ) πx ) e - ( n - 1 2 ) 2 π 2 t Imposing the IC gives v ( x, 0) = n =1 A n cos ( ( n - 1 2 ) πx )
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