Without the aid of the Wronskian, determine whether the given set of functions is linearly independent or linearly dependent on the indicated interval. a. f1(x) = ln x, f2(x) = ln x^5, (0, [infinity])
b. f1(x) = x^n, f2(x) = x^n + 2, n = 1, 2, \ldots, (−[infinity], [infinity])
c. f1(x) = x, f2(x) = x + 8, (−[infinity], [infinity])
d. f1(x) = cos(x + π), f2(x) = sin x, (−[infinity], [infinity])

Answer and Step-by-step explanation:

Solution:

(a)The function. f1(x) = ln x, f2(x) = ln x^5 is linearly independent or not.

According to “Linear independent theorem: if two functions f1and f2 are linearly dependent if one can be represent as the constant multiple of other.

f1 = n. f2

Consider given function:

f1(x) = lnx f2(x) = lnx5

(b)Set of function f1(x) = x^n, f2(x) = x^n + 2, is linearly independent or not on the interval [-∞, ∞].

According to “Linear independent theorem: if two functions f1and f2 are linearly dependent if one can be represent as the constant multiple of other.

f1 = n. f2

Consider the given function:

f1(x) = xn f2(x) = xn+2

(c)f1(x) = x, f2(x) = x + 8, (−[infinity], [infinity])

Function is not linearly independent.

Linear independent theorem states that two functions f1 and f2 are linearly dependent if one can represent as the constant multiple of other.

F1= n.f2

(d)f1(x) = cos(x + π), f2(x) = sin x. [-∞, ∞]

According to “Linear independent theorem: if two functions f1and f2 are linearly dependent if one can be represent as the constant multiple of other.

f1 = n. f2

The given function:

f1(x) = cos(x + π)f2(x)