Question

please do (iv) and explain all the steps

(4) Though I proved in class the orthogonality of eigenfunctions of the Sturm-Liouville BVP with respect to the weight function o when the Sturm-Liouville operator is regular, the orthogonality condition for eigenfunctions is true for many singular Sturm-Liouville BVP's. In this problem you will see an example. Consider then the singular Sturm-Liouville problem [(1 -u-u -1< r< 1, where u is required to be finite at ±1, meaning that limg+1 u(z) is finite. This is known as the Legendre differential equation and has eigenvalues An = n(n+ 1), n 0,1,2,... with corresponding eigenfunctions 1 dn (ar2-1)" Pn(ax) = 27 n! dan called Legendre polynomials. (iv) Calculate po, P1, P2, and p3. These are orthogonal by (iii). Multiply each by an appropriate constant to make them orthonormal.

Answer 1

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