Problem 9 - Calculus

Question

A deposit is made to a bank account paying \(8 \%\) interest compounded continuously. Payments totaling 2000 dollars per year are made from this account. (a) Write a differential equation for the balance, \(B,\) in the account after \(t\) years. (b) Find the equilibrium solution of the differential equation. Is the equilibrium stable or unstable? Explain what happens to an account that begins with slightly more money or slightly less money than the equilibrium value. (c) Write the solution to the differential equation. (d) How much is in the account after 5 years if the initial deposit is (i) \(20,000 ?\) dollars (ii) \(30,000 ?\) dollars

Step-by-Step Solution

Answer

(a) \( \frac{dB}{dt} = 0.08B - 2000 \). (b) Stable equilibrium at \( 25000 \) dollars. (c) \( B(t) = (B_0 - 25000)e^{0.08t} + 25000 \). (d) (i) \( 17409 \) dollars, (ii) \( 32459 \) dollars.

Step 1: Write the Differential Equation

The balance in the account, \( B(t) \), changes over time by gaining \( 8\% \) interest compounded continuously and losing \( 2000 \) dollars per year due to payments. The interest on the account can be represented by the term \( 0.08B(t) \), since the interest rate is \( 8\% \). The payments can be modeled as a constant withdrawal of \( 2000 \) every year. Thus, the differential equation is: \[ \frac{dB}{dt} = 0.08B(t) - 2000 \]

Step 2: Find the Equilibrium Solution

To find the equilibrium solution, set \( \frac{dB}{dt} = 0 \) and solve the equation \( 0.08B - 2000 = 0 \). Solving for \( B \) gives: \[ 0.08B = 2000 \]\[ B = \frac{2000}{0.08} = 25000 \]The equilibrium solution is \( B = 25000 \). The equilibrium is stable because if the account balance starts slightly above \( 25000 \), the withdrawals exceed the interest and the balance decreases towards \( 25000 \). If it starts slightly below \( 25000 \), the interest exceeds the withdrawals, and the balance increases towards \( 25000 \).

Step 3: Solve the Differential Equation

The differential equation \( \frac{dB}{dt} = 0.08B(t) - 2000 \) is a first-order linear differential equation. Its general solution has the form:\[ B(t) = C e^{0.08t} + 25000 \] where \( C \) is a constant determined by the initial condition \( B(0) = B_0 \). Plug \( t = 0 \) and \( B(0) = B_0 \) into the general solution to find \( C \):\[ B_0 = C + 25000 \]\[ C = B_0 - 25000 \]Thus, the solution is:\[ B(t) = (B_0 - 25000)e^{0.08t} + 25000 \]

Step 4: Compute the Balance after 5 Years with Given Initial Deposits

**(i)** For \( B(0) = 20000 \):\[ B(t) = (20000 - 25000) e^{0.08 \times 5} + 25000 \]\[ B(t) = -5000 \, e^{0.4} + 25000 \]Calculate \( e^{0.4} \approx 1.4918 \):\[ B(5) \approx -5000 \times 1.4918 + 25000 \]\[ B(5) \approx 17409.00 \]**(ii)** For \( B(0) = 30000 \):\[ B(t) = (30000 - 25000) e^{0.08 \times 5} + 25000 \]\[ B(t) = 5000 \, e^{0.4} + 25000 \]\[ B(5) \approx 5000 \times 1.4918 + 25000 \]\[ B(5) \approx 32459.00 \]

Step 5: Interpret the Results

- For an initial deposit of \( 20000 \) dollars, the account balance reduces to approximately \( 17409.00 \) dollars after 5 years.- For an initial deposit of \( 30000 \), the account balance grows to approximately \( 32459.00 \) dollars after 5 years.This illustrates the effect of the initial deposit relative to the equilibrium amount: if it's below \( 25000 \), the balance decreases, while if it's above, the balance increases.

Key Concepts

Continuous Compounding

Continuous compounding is a key concept in finance and economics, allowing for the calculation of interest that accumulates constantly over time. Unlike simple or discrete compounding, where interest is computed at specific intervals, continuous compounding assumes that interest is added to the principal at every possible moment in time. This leads to increased growth of the account balance, as interest is compounded on top of already accrued interest continuously.

Mathematically, the process of continuous compounding is modeled using the exponential function. If a principal amount grows at a continuous rate of interest, the future value of the investment after time \( t \) is given by:
\[ FV = P e^{rt} \]
where:

• \( P \) is the principal or initial amount.
• \( r \) is the annual interest rate.
• \( t \) is the time in years.

This formula shows how powerful continuous compounding can be, especially over longer periods, as the effect of compounding grows exponentially. In the context of differential equations, continuous compounding can create dynamic systems that evolve over time due to the continuous growth generated by interest.

Equilibrium Solution

In a differential equation, an equilibrium solution is a steady-state solution where the changes over time (derivatives) are zero. It indicates that the system is in balance, with no net change occurring. For the equation \( \frac{dB}{dt} = 0.08B(t) - 2000 \), finding the equilibrium involves setting \( \frac{dB}{dt} = 0 \) to see what balance \( B(t) \) remains constant.

The equilibrium solution can be found by solving the equation:
\[ 0.08B = 2000 \]
This yields \( B = 25000 \). This means that if the account balance is exactly $25,000, interest and withdrawals perfectly offset each other, maintaining the balance over time.

In practical terms, equilibrium solutions help understand whether an account will grow or shrink. If the balance is initially below the equilibrium value, accumulated interest will overpower withdrawals, causing growth toward the equilibrium. Conversely, if above, withdrawals will diminish the amount toward this stable state.

Stability Analysis

Stability analysis tells us how the solution of a system behaves when slightly perturbed. For differential equations modeling real-world phenomena, like account balances, understanding stability is crucial for anticipating system behavior.

In the context of our differential equation, the equilibrium solution \( B = 25000 \) is considered stable. To analyze stability, we examine how the system reacts to perturbations (slight changes) in initial conditions:

• If the balance \( B(t) \) starts slightly above \(25,000, withdrawals surpass interest, bringing the balance down toward \)25,000.
• Conversely, if \( B(t) \) is slightly below \(25,000, accrued interest compensates for withdrawals, nudging the balance back up to \)25,000.

This behavior reflects a stable equilibrium; perturbations naturally diminish over time, with the system returning to its original state, ensuring predictability and reliability in balance management.

Account Balance Dynamics

The dynamics of an account balance under continuous compounding and regular withdrawals can be described using differential equations. This dynamic system combines growth from interest with reductions from withdrawals.

Our core equation \( \frac{dB}{dt} = 0.08B(t) - 2000 \) captures how the balance changes continuously. The term \( 0.08B(t) \) accounts for exponential growth due to an 8% interest rate, while \(-2000 \) describes regular withdrawals.

The solution to this model, \( B(t) = (B_0 - 25000)e^{0.08t} + 25000 \), illustrates how the initial balance affects future dynamics:

• If \( B_0 < 25000 \), the balance decreases, tending toward the equilibrium value.
• If \( B_0 > 25000 \), the balance increases initially, but aims to settle at the equilibrium value.

By understanding these dynamics, we predict financial behavior over time, such as account depletion or growth, making such models invaluable in financial planning and decision-making.