Determine the values of k€R for which the following functions are linearly independent

Y1=e^(k+3)x , Y2=e^-kx , y3=e^4x

To determine the values of k€R for which the given functions are linearly independent, we need to consider their linear combination and obtain conditions on k that make them linearly dependent.

Let's assume a linear combination of the functions Y1, Y2, and Y3:
c1Y1(x) + c2Y2(x) + c3Y3(x) = 0 ----- (1)

where c1, c2, and c3 are constants.

Substituting the given functions into equation (1), we have:

c1e^(k+3)x + c2e^(-k)x + c3e^(4x) = 0 ----- (2)

To determine the values of k that satisfy equation (2) for all x, we can simplify it by factoring out the exponential terms:

e^(k+3)x(c1) + e^(-k)x(c2) + e^(4x)(c3) = 0

Since this equation must hold for all x, the coefficients of each exponential term must be zero:

c1e^(k+3) = 0 ----- (3)
c2e^(-k) = 0 ----- (4)
c3e^(4x) = 0 ----- (5)

Equations (3) and (4) imply that c1 = 0 and c2 = 0, respectively, which means that the first two functions Y1 and Y2 are linearly independent for any value of k.

For equation (5), we have c3e^(4x) = 0. Since e^(4x) ≠ 0, it implies that c3 = 0. This means that the third function Y3 is also linearly independent for any value of k.

In conclusion, the given functions Y1, Y2, and Y3 are always linearly independent for any value of k€R.