Without the aid of the Wronskian, determine whether the
Chapter , Problem 12RP(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) \(f_{1}(x)=\ln x, f_{2}(x)=\ln x^{2},(0, \quad)\)
(b) \(f_{1}(x)=x^{n}, f_{2}(x)=x^{n+1}, n=1,2, \ldots,(-\infty,)\)
(c) \(f_{1}(x)=x, f_{2}(x)=x+1,(-\infty, \quad)\)
(d) \(f_{1}(x)=\cos \left(x+\frac{\pi}{2}\right), f_{2}(x)=\sin x,(-\infty)\)
(e) \(f_{1}(x)=0, f_{2}(x)=x,(-5,5)\)
(f) \(f_{1}(x)=2, f_{2}(x)=2 x,(-\infty,)\)
(g) \(f_{1}(x)=x^{2}, f_{2}(x)=1-x^{2}, f_{3}(x)=2+x^{2},(-\infty, \quad)\)
(h) \(\begin{array}{l}f_{1}(x)=x e^{x+1}, f_{2}(x)=(4 x-5) e^{x} \\
f_{3}(x)=x e^{x},(-\infty, \quad)\end{array}\)
Text Transcription:
f_1(x)=lnx, f_2(x)=ln x^2,(0,)
f_1(x)=x^n, f_2(x)=x^n+1, n=1,2,(-\infty,)
f_1(x)=x, f_2(x)=x+1,(-\infty,)
f_1(x)=\cos(x+fracpi2t), f_2(x)=\sin x,(-\infty)
f_1(x)=0, f_2(x)=x,(-5,5)
f_1(x)=2, f_2(x)=2 x,(-\infty,)
f_1(x)=x^2, f_2(x)=1-x^2,f_3(x)=2+x^2,(-\infty,)
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