Someone is asked whether and are linearly independent on

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

Mathematical Methods for Physics, Ch Higher-order linear equations, the Wronskian and variation of parameters

Problem

Someone is asked whether y1(t)=ty_1(t) = t and y2(t)=t2y_2(t) = t^2 are linearly independent on −1≤t≤1-1 \le t \le 1. They compute the Wronskian at t=0t = 0: W(y1,y2)(0)=y1(0)y2′(0)−y2(0)y1′(0)=0⋅0−0⋅1=0.W(y_1, y_2)(0) = y_1(0)y_2'(0) - y_2(0)y_1'(0) = 0\cdot 0 - 0\cdot 1 = 0. They conclude: "the Wronskian vanishes, so y1y_1 and y2y_2 are linearly dependent on this interval." ((a)) Compute W(y1,y2)(t)W(y_1, y_2)(t) for general tt, not just at t=0t = 0. ((b)) Say whether the conclusion is correct, and explain the flaw in the reasoning. ((c)) Both y1y_1 and y2y_2 solve t2y′′−2ty′+2y=0t^2y'' - 2ty' + 2y = 0. State why the all-or-nothing property proved from Abel's identity for solutions with continuous coefficients does not rule out an isolated zero of WW here.

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