Second, even if we were comfortable with complex-value functions, in this course we do not address the idea of a derivative for such functions. First, these functions take on complex (imaginary) values, and a complete discussion of such functions is beyond the scope of this text. However, there are problems with this approach. If we follow the same process we used for distinct real roots-using the roots of the characteristic equation as the coefficients in the exponents of exponential functions-we get the functions e ( α + β i ) x e ( α + β i ) x and e ( α − β i ) x e ( α − β i ) x as our solutions. This creates a little bit of a problem for us. Thus, we see that when b 2 − 4 a c < 0, b 2 − 4 a c < 0, the roots of our characteristic equation are always complex conjugates. The complex number α + β i α + β i is called the conjugate of α − β i. We must use the imaginary number i = −1 i = −1 to find the roots, which take the form λ 1 = α + β i λ 1 = α + β i and λ 2 = α − β i. In this case, when we apply the quadratic formula, we are taking the square root of a negative number. The third case we must consider is when b 2 − 4 a c < 0. Then, there are constants, c 1 c 1 and c 2, c 2, not both zero, such that Assume f 1 ( x ) f 1 ( x ) and f 2 ( x ) f 2 ( x ) are linearly independent. Next, we show that if two functions are linearly dependent, then either one is identically zero or they are constant multiples of one another. If, on the other hand, neither f 1 ( x ) f 1 ( x ) nor f 2 ( x ) f 2 ( x ) is identically zero, but f 1 ( x ) = C f 2 ( x ) f 1 ( x ) = C f 2 ( x ) for some constant C, C, then choose c 1 = 1 C c 1 = 1 C and c 2 = −1, c 2 = −1, and again, the condition is satisfied. If one of the functions is identically zero-say, f 2 ( x ) ≡ 0 f 2 ( x ) ≡ 0-then choose c 1 = 0 c 1 = 0 and c 2 = 1, c 2 = 1, and the condition for linear dependence is satisfied. From a practical perspective, we see that two functions are linearly dependent if either one of them is identically zero or if they are constant multiples of each other.įirst we show that if the functions meet the conditions given previously, then they are linearly dependent. In this chapter, we usually test sets of only two functions for linear independence, which allows us to simplify this definition. A set of functions that is not linearly dependent is said to be linearly independent. A set of functions f 1 ( x ), f 2 ( x ),…, f n ( x ) f 1 ( x ), f 2 ( x ),…, f n ( x ) is said to be linearly dependent if there are constants c 1, c 2 ,… c n, c 1, c 2 ,… c n, not all zero, such that c 1 f 1 ( x ) + c 2 f 2 ( x ) + ⋯ + c n f n ( x ) = 0 c 1 f 1 ( x ) + c 2 f 2 ( x ) + ⋯ + c n f n ( x ) = 0 for all x over the interval of interest.
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