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Unformatted text preview: Singular ; it is a Sturm-Liouville problem because it can be written in the form ( ( x- 2) y ′ ) ′ + y + λy = 0 , which is S-L with p ( x ) = x- 2, q ( x ) = 1 and σ ( x ) = 1. But it is singular because p (2) = 0. 4. The problem is y ′′- y ′ + λy = 0 , < x < 2 , y ′ (0) = 0 , y (2)- y ′ (2) = 0 . This problem does nt look like an S-L problem at first glance, but it might be rewritten so as to be an S-L problem. Is there a function p ( x ) such that p ( x ) > 0 in [0 , 2] and multiplying the differential equation by p ( x ) it is of the form ( py ′ ) ′ + qy + λσy = 0 for some q, σ ?. We want these two equations to be the same, for some function p : = p ( y ′′- y ′ + λy ) = py ′′- py ′ + λpy, = ( py ′ ) ′ + qy + λσy = py ′′ + p ′ y ′ + qy + λσy. Clearly (I hope) we need (and it suﬃces) p such that p ′ =- p . Solving this differential equation, we get p ( x ) = Ce − x , for constant C . We can take C = 1. We conclude: The differential equation is equivalent to d dx ( e − x y ) + λe − x y = 0 , so that we have a regular S-L problem, with p ( x ) = e − x , q ( x ) = 0 , σ ( x ) = e − x , κ 1 = 0 , κ 2 = 1 , κ 3 = 1 , κ 4 =- 1 . 2 5. The problem is 2 y ′′ + y ′ + ( λ + x ) y = 0 , < x < ∞ , y (0) = 0 , y ( x ) , y ′ ( x ) bounded as x → ∞ . Solution. The only question here is whether this is an S-L problem; if it is, it is singular because the interval is infinite among other reasons. To see if it is S-L, we must see if there is a functioninterval is infinite among other reasons....
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- Spring '13
- Boundary value problem, Sturm–Liouville theory, S-L problem, qy + λσy