Consider the purely mathematical

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations

Conditions of Validity, Ch Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise

Problem

Consider the purely mathematical, dimensionless equation (x2−4) y′′+2x y′+ln⁡xx−3 y=0.(x^2-4)\,y'' + 2x\,y' + \frac{\ln x}{x-3}\,y = 0. ((a)) With y(1)y(1) and y′(1)y'(1) given, find the largest open interval on which the existence and uniqueness theorem guarantees a unique solution. ((b)) Do the same with the data given at x0=52x_0=\tfrac52 instead. ((c)) Two solutions y1,y2y_1,y_2 on the interval of part (a) have Wronskian W(1)=1W(1)=1. Find W(12)W(\tfrac12), and explain why WW cannot vanish anywhere on that interval.

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