Question

#1 part a and b please write solutions neatly

1. (16 points) Determine whether the given set of functions is linearly dependent or linearly independent on the indicated interval. Justify your answers. (a) (8 points) f(x) = x + 2 cos?x, f?(x) = 3sin?x, f(x) = x+2 on (-0,0). (b) (8 points) f (x) = ed" and 12(x) = e 4s are solutions of the linear homogeneous differential equation y" + y' - 12y = 0 on (-0,0).

Answer 1

solution 1.) a.) Given that, filx) = 8 + 2cu23 f(x) = Esinex f} (x) = X+2 if these funeHions are linearly dependent then We com white them in the following manner-' Crfy(x) + Cafq[) +Gef,CM) =0 where Gp Cg and Co not all equal to 2000. C/Cx+ 2cos2x) + C2 (3)ừn2x) +63(x+2) = 0 Q1X + 2C, Cos?X + 362 3in284 +6gX+262 = 0 6X + 26 (852x +342 (91WX) H+69X+262 = 0

G1X +QC, (092% +3C951mPx +692 +263=0 On observing, we can see G=-C3 26 = 362 26 = 263 We get C=3 x3 =2 G=-C3 Cz=-3 Thus, at C = 3, C2 = 2 and C3=-3 4f, +Caf, +(389 = 0 Thus, the set of given functions is linearly dependent

b.) Given fi(x) = (x , f (x) = (-42 If these set functions are linearly dependent then we can write, G e34 + 6 e 4x=0 But this is not posible as ex porential functions are never 2010 and there has to be a zero scalar to multiply with them to make them zero. This is possible only if G = C = 0 Thus, the given set of functions in linearly independent. p3x and e-lux are linearly independent
In part a.) The set of functions are linearly dependent while in part b) the set of functions are linearly independent .