## A Regular Sturm-Liouville Problem### Problem Statement:Determine the eigenvalues (\(\lambda_n\)) and eigenfunctions (\(\varphi_n(x)\)) for the differential equation:\[\varphi''(x) - 2x\varphi'(x) + (\lambda + x^2)\varphi(x) = 0\]for \(0 \leq x \leq 1\), along with the boundary conditions:\[\varphi'(0) = \varphi(1) = 0.\]**Hint:** See Problem #4 of Homework #4.### Instructions:Determine the "dot" product for which \(\varphi_m \cdot \varphi_n = 0\) when \(m \neq n\), and use this to determine the coefficients \(a_n\) if the function \[f(x) = e^{\frac{1}{2}x^2}\]is expanded as\[f(x) = e^{\frac{1}{2}x^2} = \sum_{n=1}^{\infty} a_n \varphi_n(x)\]for \(0 \leq x \leq 1\).

Answered: Determine the eigenvalues (2„) and…

Determine the eigenvalues (2„) and eigenfunctions (9„(x)) for the differential equation p" (x) – 2xo'(x) + (2 + x²)@(x) = 0 %3D for 0

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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