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Differential Equations 2, 2021
Consider bounded solutions of the eigenvalue problem\[\tag{*}ℒy(x)≡\left(1-x^2\right) y''(x)-2 x y'(x)=λy(x), \quad-1<x<1\](a) Use the inner product relation to compute $ℒ ^*$ and show that the boundary terms vanish identically. Why are no boundary conditions given for (*)?
(b) Convert (*) to Sturm–Liouville form. What orthogonality relation do the eigenfunctions satisfy?
(c) Verify that $y_0(x)=1$ is an eigenfunction for $λ_0=0$. For the inhomogeneous problem $ℒy(x)=f(x)$ to be solvable for $y$, what condition must $f$ satisfy?
(d) Consider the equation $ℒy(x)=-2x$. Explain via the Fredholm Alternative why this problem should have a non-unique solution. Show that\[y=x+A \log \left(\frac{1+x}{1-x}\right)+B\]is a solution for any values of $A$ and $B$. What can you conclude about the constant $A$?
4.7.3 Singular SL problems |
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