The exponential function y (x) = ce^ax satisfies the conditions y (0) = 2 and y (1) = 1. find the constraint of c and

a. what is y (2) ?

The conditions y (0) = 2 and y (1) = 1 are initial value conditions. It means that when x = 0, y=2 and when x=1, y=1. From these two conditions, we can formulate two equations which is enough to solve the two unknowns, c and a.

y (0) = 2

2 = ceᵃ ⁽⁰⁾ = ce⁰ = c (1)

Thus, c=2

y (1) = 1

1 = 2eᵃ ⁽¹⁾ = 2eᵃ

eᵃ = 1/2

ln eᵃ = ln (1/2)

a = ln (1/2) = - 0.693

Thus, the general equation is:

y = 2e⁻⁰⁶⁹³ ⁽ˣ⁾

So, when x = 2,

y (2) = 2e⁻⁰⁶⁹³ ⁽²⁾ = 1/2