Chem Differential Eq HW Solutions Fall 2011 96

Chem Differential Eq HW Solutions Fall 2011 96 - 96 Chapter...

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96 Chapter 6 Sturm-Liouville Theory with Engineering Applications Solutions to Exercises 6.2 1. Sturm-Liouville form: ( xy ) + λy = 0, p ( x ) = x , q ( x ) = 0, r ( x ) = 1. Singular problem because p ( x ) = 0 at x = 0. 5. Divide the equation through by x 2 and get y x - y x 2 + λ y x = 0. Sturm-Liouville form: ( 1 x y ) + λ y x = 0, p ( x ) = 1 x , q ( x ) = 0, r ( x ) = 1 x . Singular problem because p ( x ) and r ( x ) are not continuous at x = 0. 9. Sturm-Liouville form: ( (1 - x 2 ) y ) + λy = 0, p ( x ) = 1 - x 2 , q ( x ) = 0, r ( x ) = 1. Singular problem because p ( ± 1) = 0. 13. Before we proceed with the solution, we can use our knowledge of Fourier series to guess a family of orthogonal functions that satisfy the Sturm-Liouville problem: y k ( x ) = sin 2 k +1 2 x , k = 0 , 1 , 2 ,... . It is straightforward to check the validity of our guess. Let us instead proceed to derive these solutions. We organize our solution after Example 2. The differential equation fits the form of (1) with p ( x ) = 1, q ( x ) = 0, and r ( x ) = 1. In the boundary conditions, a = 0 and b = π , with c 1 = d 2 = 1 and c 2 = d 1 = 0, so this is a regular Sturm-Liouville problem.
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