Question

step by step please

Without the aid of the Wronskian, determine whether the given set of functions is linearly independent or linearly dependent on the indicated interval. (a) (%) = In x, f(x) = Inx",(0, ) linearly dependent linearly independent (b) (x) = 2x",,(*) - +, n = 1, 2, Vdots, (-,) linearly dependent linearly independent (c) (X) = x, f(x) - X+4.(-.,-) linearly dependent linearly independent ,) (d) {(x) = cos(x + 2), f(x) - sinx, ( linearly dependent linearly independent (e) 0) = 0, 1,(X) = x (-10,10) linearly dependent linearly independent ( (*) - 9,1(%) - 9x, (,-) linearly dependent linearly independent (x) = 1 + x?(- -) (9) 7,0) = x2,104) - 4 - x2 linearly dependent linearly independent (1) f(x) xe" (-m, am) xe" (X) = (2x - 5), (8) linearly dependent linearly independent

Answer 1

Answen (a) considen ,(x) in x, a(x) n Since 2()An : S An x 5x) So, 2(a)- 51() 0 Hence, the Junction lin caydependen (b Conside ,()2x, ) n1,2(- ) Since t2(x): A Hen e, ) com not be expaessed as cons tant mutipie o 4 tunction lincay inde pendent Hence, the Considen the unction,(x)x, 2() xt,C-) Since t is nota constant mutipe o4 , the tunchion s e lincaay independent Hence the unchtons ae tineay inde Penden

Consider the unchion (x): CoS(x), 2(x) SIn since ) Cos (t - -COSX incay independent Hence,the unchions Oue Consider the untions (1) 0, 2() Since x) O : DX DX2() ane linca dependen The unctions 3,) 9 1)9x, (-, Considen the tunchons Since 20x) 9x 2C) ,x) The Jun chion 2(x) a not be expessed as constan Hence the un chions oae lin Co independen- muli pie o ) (h Considen the Junchion ,(x): xe 2) (ST-s)e )e t 3 The sum o 3 Junchions can not be equai to D the tunction Hence ae linecny independent