Question

The government of the Republic of Lemon Island plans to transform their lemon-based

economy into a tourism-based economy. They develop the following model to predict the
long-term effect of this transformation for the size of their economy y:
dy
dt = (A − Be−t/5
)y; A, B > 0.
(a) Find the general solution y(t) for this equation.
(b) The Republic of Lemon Island has a current economic growth of 2% per year and
the government expects the transformation to cause this growth rate to triple in
the long term. Explain why values A = 0.06 and B = 0.04 are a good choice to
approximate this behaviour.
(c) Using A = 0.06 and B = 0.04, find the particular solution y(t) assuming that the
size of the economy of Lemon Island before the transformation is $50 000 000, and
that the transformation causes the size of the economy to shrink by $10 000 000 at
t = 0.
(d) The size of the economy of Lemon Island without the transformation can be mod-
eled using the same equation with parameters A = 0.02 and B = 0.
In which year after the transformation will the size of the transformed economy
surpass the size of the economy without the transformation?
(e) Elections on Lemon Island are in five years time. The Lemon Island Polling In-
stitute (LIPI) predicts that the current government will only be re-elected, if the
growth rate of the economy is, by then, at least 4%. According to the first model,
can the current government expect to be re-elected?

Answer 1

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.