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
