Step-by-step explanation:
dy/dt = (A − Be^(-t/5)) y
(a) Find the general solution by separating the variables and integrating.
dy / y = (A − Be^(-t/5)) dt
dy / y = [A + 5B (-⅕ e^(-t/5))] dt
ln |y| = At + 5B e^(-t/5) + C
y = e^(At + 5B e^(-t/5) + C)
y = Ce^(At + 5B e^(-t/5))
(b) As t approaches infinity, dy/dt approaches Ay. The rate of growth, A, is triple the initial rate of growth 0.02, so A = 0.06.
When t = 0, dy/dt = (A − B)y. The rate of growth, A−B, is 0.02. We know A = 0.06, so B = 0.04.
(c) Given that A = 0.06, B = 0.04, and y(0) = 50×10⁶ − 10×10⁶ = 40×10⁶:
40×10⁶ = Ce^(0.06(0) + 5(0.04) e^(-0/5))
40×10⁶ = Ce^(0.2)
C = 40×10⁶ e^(-0.2)
y = 40×10⁶ e^(-0.2) e^(0.06t + 0.2 e^(-t/5))
y = 40×10⁶ e^(-0.2 + 0.06t + 0.2 e^(-t/5))
(d) If A = 0.02 and B = 0, and there is no transformation (y(0) = 50×10⁶), then:
50×10⁶ = Ce^(0.02(0) + 0)
50×10⁶ = C
y = 50×10⁶ e^(0.02t)
Comparing to the answer from part (c):
40×10⁶ e^(-0.2 + 0.06t + 0.2 e^(-t/5)) = 50×10⁶ e^(0.02t)
e^(-0.2 + 0.04t + 0.2 e^(-t/5)) = 5/4
-0.2 + 0.04t + 0.2 e^(-t/5) = ln(5/4)
-5 + t + 5e^(-t/5) = 25 ln(5/4)
t + 5e^(-t/5) = 5 + 25 ln(5/4)
Solve with a calculator:
t = 9.886
The transformed economy surpasses the untransformed economy in the 10th year.
(e) In year t=5, the size of the transformed economy is:
y = 40×10⁶ e^(-0.2 + 0.06(5) + 0.2 e^(-5/5))
y = 47.6×10⁶
The percent growth is:
(47.6×10⁶ − 40×10⁶) / 40×10⁶ × 100% = 19%
The growth rate is greater than 4%, so the current government can expect to be reelected.
