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Unformatted text preview: To see if it is S-L, we must see if there is a function μ ( x ) such that after multiplying the differential equation by μ ( x ), the equation has the form ( py ′ ) ′ + qy + λσy = 0. In ur case, multiplication by a function μ ( x ) gives 0 = 2 μy ′′ + μy ′ + μxy + λμy. We want this to be 0 = py ′′ + p ′ y + qy + λσy = 0 . Thus 2 μ = p , μ = p ′ , q = μx , σ = μ . From the first two equations, p ′ = 1 2 p ; a non zero solution of this ODE is p ( x ) = e x/ 2 . Then μ = 1 2 e x/ 2 . The equation can be rewritten in the form d dx ( e x/ 2 y ′ ) + 1 2 xe x/ 2 y + 1 2 xe x/ 2 λy = 0 , an S-L type equation with p ( x ) = e x/ 2 , q ( x ) = 1 2 xe x/ 2 , σ ( x ) = 1 2 e x/ 2 . 6. The problem is x 2 y ′′ + xy ′ + (2 λ- 1) y = 0 , < x < 1 , y ( x ) , y ′ ( x ) bounded as x → 0+ , y (1)- y ′ (1) . Solution. Once again, the only question here is whether this is an S-L problem; if it is, it is singular because the boundary condition at 0 is not of the regular type and, more importantly, while p ( x ) is not x 2 it will still turn out to be 0 at 0. If one proceeds as in the previous problem, one finds that the function μ by which one should multiply the ODE is μ ( x ) = 1 /x , after which the ODE can be rewritten in the form ( xy ′ ) ′- 1 x y + 2 λy = 0 , an S-L type equation with p ( x ) = x , q ( x ) =- 1 x , σ ( x ) = 2. 7. This was the Quiz 2 problem: Compute the eigenvalues and eigenfunctions of the following regular S-L problem: y ′′ + λy = 0 , < x < π y (0) = 0 , y ′ ( π ) = 0 ....
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- Spring '13
- Boundary value problem, Sturm–Liouville theory, S-L problem, qy + λσy