Question

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).

Answer 1

a = -0.6931471806

C = 2

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