The exponential function y (x) = Ceαx satisfies the conditions y (0) = 2 and y (1) = 1. (a) Find the constants C and α. Enter the exact value of α. Enclose arguments of functions, numerators, and denominators in parentheses. For example, sin (2x) or (a-b) / (1+n).

a = - 0.6931471806

C = 2

Step-by-step explanation:

Given that y (x) = Ce^ (ax) satisfies the conditions y (0) = 2, and y (1) = 1.

We need to find the values of a and C.

To do this, we are going to apply the conditions given to the function y (x), this way, we will obtain two equations which can then be solved simultaneously.

Applying y (0) = 2, put y (x) = 2 and x = 0 in y (x) = Ce^ (ax)

2 = Ce^ (a * 0)

C = 2 (because e^0 = 1) ... (1)

Again, applying y (1) = 1, put y = 1 and x = 1 in y (x) = Ce^ (ax)

1 = Ce^ (a * 1)

Ce^a = 1 ... (2)

From (1), C = 2. Putting this in (2), we have

2e^a = 1

Divide both sides by 2

e^a = 1/2

This equation has an index which could make finding a prove difficult by a direct mean, to find a, we need to take logarithm of both sides,

log (e^a) = log (1/2)

(a) log (e) = log (1/2)

a = log (1/2) / (log e)

a = (-0.3010299957) / (0.4342944819)

a = - 0.6931471806