Solved: Without the aid of the Wronskian determine whether the given set of functions is
Chapter 3, Problem 10(choose chapter or problem)
Without the aid of the Wronskian determine whether the given set of functions is linearly independent or linearly dependent on the indicated interval.
(a) \(f_{1}(x)=\ln x, f_{2}(x)=\ln x^{2}\), \((0, \infty)\)
(b) \(f_{1}(x)=x^{n}, f_{2}(x)=x^{n+1}\), \(n=1,2, \ldots,(-\infty, \infty)\)
(c) \(f_{1}(x)=x, f_{2}(x)=x+1\), \((-\infty, \infty)\)
(d) \(f_{1}(x)=\cos (x+\pi / 2), f_{2}(x)=\sin x\), \((-\infty, \infty)\)
(e) \(f_{1}(x)=0, f_{2}(x)=x\), \((-5,5)\)
(f) \(f_{1}(x)=2, f_{2}(x)=2 x\), \((-\infty, \infty)\)
(g) \(f_{1}(x)=x^{2}, f_{2}(x)=1-x^{2}, f_{3}(x)=2+x^{2}\), \((-\infty, \infty)\)
(h) \(f_{1}(x)=x e^{x+1}, f_{2}(x)=(4 x-5) e^{x}, f_{3}(x)=x e^{x}\), \((-\infty, \infty)\)
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