Question

y_1(t)=costsint, y_2(t)=sin2t
Use definition 1 to determine whether fxn's y1 and y2 are linearly dependant on interval (0,1)

Answer 1

Definition 1 is ... A pair of fxns y1(t) and y2(t) is said to be linearly independent on interval I if and only if neither of them is a constant multiple of the other on all of I. We say y1 and y2 are linearly dep on I if one of them is a constant multiple of the other on all of I

Answer 2

sin (2t) = 2sin(t) cos(t) soooo

Answer 3

its linear dependent but why?
what about y1=e^3t y2=e^-4t
or y1=te^2t y2=e^2t both of these are linear independent but it seems like they are multiples of the other

Answer 4

well, we can either setup a matrix and define the determinant as zero .... or we can recall from trig classes that the identity for sin(2t) is 2sin(t) cos(t)

Answer 5

matrix-wise

f(x) g(x)
f'(x) g'(x)

fg' - gf' = 0 .... they are dependant

Answer 6

thanks @amistre64 ill try the matrix for the other two...ive also never taken trig so it makes calc and diffy q a little tougher than it should be for me

Answer 7

:) good luck

Answer 8

gracias!