2017 Exam 2.pdf - Math 3363 Examination II Fall 2017 Please use a pencil and do the problems in the order in which they are listed You may use the

2017 Exam 2.pdf - Math 3363 Examination II Fall 2017 Please...

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Math 3363 Examination II Fall 2017 Please use a pencil and do the problems in the order in which they are listed . You may use the following information without derivation. A proper listing of eigenvalues and eigenfunctions for (i) 00 ( ) =  ( ) for 0 , (ii) (0) = 0 , and (iii) ( ) = 0 is { } =1 and { } =1 where = (  ) 2 and ( ) = sin  A proper listing of eigenvalues and eigenfunctions for (i) 00 ( ) =  ( ) for 0 , (ii) 0 (0) = 0 , and (iii) 0 ( ) = 0 is { } =0 and { } =0 where = (  ) 2 and ( ) = cos  Note that 0 = 0 and 0 ( ) = 1 A proper listing of eigenvalues and eigenfunctions for (i) 00 ( ) =  ( ) for 0 , (ii) (0) = 0 , and (iii) 0 ( ) = 0 is { } =1 and { } =1 where = μ (2 1) 2 2 and ( ) = sin (2 1)  2 1
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A proper listing of eigenvalues and eigenfunctions for (i) 00 ( ) =  ( ) for 0 , (ii) 0 (0) = 0 , and (iii) ( ) = 0 is { } =1 and { } =1 where = μ (2 1) 2 2 and ( ) = cos (2 1)  2 1. Suppose that each of and is a positive number, is a number, 00 ( ) =  ( ) for 0  0 ( ) +  ( ) = 0 for 0 and (   ) = ( ) ( ) for 0 and 0 Show that   (   ) = 2  2 (   ) for 0 and 0 Solution.   (   ) = ( ) 0 ( ) = ( )(  ( )) =  ( ) ( ) and 2  2 (   ) =  00 ( ) ( ) = (  ( )) ( ) =  ( ) ( ) for 0 and 0 . Thus   (   ) = 2  2 (   ) for 0 and 0 2. Suppose that each of and is a positive number. Find the function that satis fi es 00 ( ) =  ( ) for 0 , ( ) = 0 , and (0) = 1 . This is not an eigenvalue problem. Solution. is a solution to the di ff erential equation if and only if ( ) = 1 sinh  + 2 sinh ( ) 2
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for some pair of numbers 1 and 2 0 = 1 sinh  + 2 sinh ( ) = 1 sinh  so 1 = 0 Then 1 = 2 sinh ( 0) = 2 sinh  so 2 = 1 sinh  Thus ( ) = sinh ( ) sinh  3. Suppose that   0 ,   0 , is a function de fi ned on [0   ] × [0   ] , and is a function de fi ned on the boundary of [0   ] × [0   ] . Show that there is at most one function such that is continuous on [0   ] × [0   ] , 2  2 (   ) + 2  2 (   ) = 0 for all (   ) in the interior of [0   ] × [0   ] , and (   ) = (   ) for all (   ) on the boundary of [0   ] × [0   ] . Do not use series solutions.
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