Chem Differential Eq HW Solutions Fall 2011 101

Chem Differential Eq HW Solutions Fall 2011 101 - Section...

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Section 6.4 Fourth Order Sturm-Liouville Theory 101 Exercises 6.4 1. This is a special case of Example 1 with L =2and λ = α 4 . The values of α are the positive roots of the equation cos 2 α = 1 cosh 2 α . There are inFnitely many roots, α n ( n =1 , 2 ,... ), that can be approximated with the help of a computer (see ±igure 1). To each α n corresponds one eigenfunction X n ( x ) = cosh α n x - cos α n x - cosh 2 α n - cos 2 α n sinh 2 α n - sin 2 α n (sinh α n x - sin α n x ) . 5. There are inFnitely many eigenvalues λ = α 4 , where α is a positive root of the equation cos α = 1 cosh α . As in Example 1, the roots of this equation, α n ( n , 2 ), can be approxi- mated with the help of a computer (see ±igure 1). To each α n corresponds one eigenfunction X n ( x ) = cosh α n x - cos α n x - cosh α n - cos α n sinh α n - sin α n (sinh α n x - sin α n x ) . The eigenfunction expansion of f ( x )= x (1 - x ), 0 <x< 1, is f ( x ± n =1 A n X n ( x ) , where A n == ² 1 0 x (1 - x ) X n ( x ) dx ² 1 0 X 2 n ( x ) dx . After computing several of these coefficients, it was observed that: ³ 1 0 X 2 n ( x ) dx = 1 for all n , 2 ,..., A 2 n = 0 for all n , 2 ,.... The Frst three nonzero coefficients are
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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