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Chapter 11: Problem 6

A bank account that earns \(10 \%\) interest compounded continuously has an initial balance of zero. Money is deposited into the account at a constant rate of 1000 dollars per year. (a) Write a differential equation that describes the rate of change of the balance \(B=f(t)\). (b) Solve the differential equation to find the balance as a function of time.

Short Answer

Expert verified
The differential equation is \( \frac{dB}{dt} = 0.1B + 1000 \) and the balance is \( B(t) = 10000(e^{0.1t} - 1) \).

Step by step solution

01

Identify the type of differential equation

The differential equation will involve the rate of change of the balance in the account. Since the interest is compounded continuously, we use the formula for continuously compounding interest, and additionally, a constant deposit rate is involved.
02

Formulate the differential equation

Let the balance of the account at time \( t \) be \( B(t) \). The rate of change of \( B(t) \) is given by the sum of the interest earned continuously, which is \( 0.1B(t) \), and the constant deposit rate, 1000. Thus, the differential equation becomes: \[ \frac{dB}{dt} = 0.1B + 1000. \]
03

Solve the differential equation

This is a first-order linear differential equation of the form \( \frac{dB}{dt} - 0.1B = 1000 \). Use the integrating factor method to solve it. The integrating factor is \( e^{-0.1t} \). Multiply through by the integrating factor: \[ e^{-0.1t}\frac{dB}{dt} - 0.1e^{-0.1t}B = 1000e^{-0.1t}. \]
04

Simplify and integrate

The left side of the equation is the derivative of \( e^{-0.1t}B \). Integrate both sides with respect to \( t \): \[ \int \frac{d}{dt}(e^{-0.1t}B) \, dt = \int 1000e^{-0.1t} \, dt. \] Thus, \[ e^{-0.1t}B = -10000e^{-0.1t} + C. \]
05

Solve for B(t)

Isolate \( B(t) \) by multiplying through by \( e^{0.1t} \): \[ B(t) = -10000 + Ce^{0.1t}. \] Since the initial balance is zero \( B(0) = 0 \), substitute \( t = 0 \) into the equation: \[ 0 = -10000 + C. \] Therefore, \( C = 10000 \).
06

Final expression for B(t)

Substitute \( C = 10000 \) back into the expression for \( B(t) \): \[ B(t) = -10000 + 10000e^{0.1t}. \] Simplify to: \[ B(t) = 10000(e^{0.1t} - 1). \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Compounding Interest
Continuous compounding interest is a fascinating concept in finance. It assumes that interest is calculated and added to the principal balance at every moment, continuously.
This method results in a little more interest accrued than with traditional compounding methods.

In continuous compounding, the growth of your investment is exponential. The mathematical model for continuous compounding is inspired by natural exponential growth and can be expressed by the formula:
  • \( A = Pe^{rt} \)
where:
  • \( A \) is the final amount
  • \( P \) is the principal balance
  • \( r \) is the annual interest rate
  • \( t \) is the time in years.

By using this formula, you can see how investments grow over time when interest is added continuously.
First-order Linear Differential Equation
A first-order linear differential equation is a type of equation that involves the derivatives of a function with respect to one variable, like time. In our context, it represents how the balance changes in a bank account.
The general form is:
  • \( \frac{dy}{dt} + Py = Q \)
where:
  • \( y \) is the unknown function
  • \( P \) and \( Q \) are functions of \( t \)

These equations model many real-world situations, often pertaining to rates and growth processes.

In the exercise, the equation for the bank balance was:
  • \( \frac{dB}{dt} = 0.1B + 1000 \)
This equation incorporates continuous compounding interest and constant deposits.
Integrating Factor Method
The integrating factor method is a clever technique used to solve first-order linear differential equations. It's invaluable when the differential equation isn't readily separable.
The core idea is to multiply the entire differential equation by an integrating factor, which greatly simplifies the equation.

For our equation \( \frac{dB}{dt} - 0.1B = 1000 \), the integrating factor is \( e^{-0.1t} \).
  • When the equation is multiplied by this factor, the left-hand side becomes the derivative of a product.
This clever manipulation allows the equation to be integrated straightforwardly.

After integration, constants are adjusted based on initial conditions, such as the initial balance of zero, to find the specific solution.
Rate of Change
The rate of change is a fundamental concept in calculus and describes how a quantity changes over time. In a differential equation, it is represented by a derivative, like \( \frac{dB}{dt} \) in our example.
This represents how quickly or slowly the bank balance \( B \) changes with respect to time \( t \).

In the given problem, the rate of the balance change is affected by two factors:
  • Interest accruing continuously due to the 10% rate
  • The constant deposit of 1000 dollars per year.
Together, these create a dynamic interplay that causes the balance to grow exponentially over time, showcasing the beauty and power of differential equations in modeling real-world financial scenarios.

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Most popular questions from this chapter

A bank account earns \(5 \%\) annual interest, compounded continuously. Money is deposited in a continuous cash flow at a rate of 1200 dollars per year into the account. (a) Write a differential equation that describes the rate at which the balance \(B=f(t)\) is changing. (b) Solve the differential equation given an initial balance \(B_{0}=0\) (c) Find the balance after 5 years.

(a) Define the variables. (b) Write a differential equation to describe the relationship. (c) Solve the differential equation. In \(2010,\) the population of India was 1.15 billion people and increasing at a rate proportional to its population. If the population is measured in billions of people and time is measured in years, the constant of proportionality is 0.0135.

(a) An object is placed in a \(68^{\circ} \mathrm{F}\) room. Write a differential equation for \(H,\) the temperature of the object at time \(t.\) (b) Find the equilibrium solution to the differential equation. Determine from the differential equation whether the equilibrium is stable or unstable. (c) Give the general solution for the differential equation. (d) The temperature of the object is \(40^{\circ} \mathrm{F}\) initially and \(48^{\circ} \mathrm{F}\) one hour later. Find the temperature of the object after 3 hours.

Are the statements true or false? Give an explanation for your answer. The system of differential equations \(d x / d t=-x+x y^{2}\) and \(d y / d t=y-x^{2} y\) requires initial conditions for both \(x(0)\) and \(y(0)\) to determine a unique solution.

Consider a conflict between two armies of \(x\) and \(y\) soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and \(t\) represents time since the start of the battle, then \(x\) and \(y\) obey the differential equations $$\begin{array}{l} \frac{d x}{d t}=-a y \\ \frac{d y}{d t}=-b x \quad a, b>0 \end{array}$$. Near the end of World War II a fierce battle took place between US and Japanese troops over the island of Iwo Jima, off the coast of Japan. Applying Lanchester's analysis to this battle, with \(x\) representing the number of US troops and \(y\) the number of Japanese troops, it has been estimated \(^{31}\) that \(a=0.05\) and \(b=0.01\) (a) Using these values for \(a\) and \(b\) and ignoring reinforcements, write a differential equation involving \(d y / d x\) and sketch its slope field. (b) Assuming that the initial strength of the US forces was 54,000 and that of the Japanese was 21,500 draw the trajectory which describes the battle. What outcome is predicted? (That is, which side do the differential equations predict will win?) (c) Would knowing that the US in fact had 19,000 reinforcements, while the Japanese had none, alter the outcome predicted?
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