Let y(t) represent your bank account balance, in dollars, after t years. Suppose you start with $30000 in the account. Each year the account earns 3% interest, and you withdraw $3000. Write the differential equation modeling this situation. dy dt y(0) = %3D
Let y(t) represent your bank account balance, in dollars, after t years. Suppose you start with $30000 in the account. Each year the account earns 3% interest, and you withdraw $3000. Write the differential equation modeling this situation. dy dt y(0) = %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Modeling Differential Equations in Financial Situations**
In this example, we will model the balance of a bank account over time using a differential equation.
**Problem Statement:**
Let \( y(t) \) represent your bank account balance, in dollars, after \( t \) years. Suppose you start with $30,000 in the account. Each year the account earns 3% interest, and you withdraw $3,000 annually.
**Objective:**
Write the differential equation that models this situation.
**Solution:**
Start by identifying all the components of the given problem:
1. Initial balance: $30,000
2. Annual interest rate: 3% (or 0.03 as a decimal)
3. Annual withdrawal: $3,000
The balance change is affected by both the interest earned and the withdrawals. The interest contributes \( 0.03 \times y(t) \) dollars each year, and the withdrawal subtracts $3,000 each year.
The rate of change of the balance, \( \frac{dy}{dt} \), can be expressed by the differential equation:
\[ \frac{dy}{dt} = 0.03y - 3000 \]
Given that the initial balance is $30,000, we have:
\[ y(0) = 30000 \]
**Summary:**
The differential equation model for the bank account balance is:
\[ \frac{dy}{dt} = 0.03y - 3000 \]
With the initial condition:
\[ y(0) = 30000 \]
This system provides a mathematical framework for understanding how your bank account balance changes over time considering interest and withdrawals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b092f3-63db-4d93-982f-67d0473d8e68%2F826e2123-5bab-498c-8548-6ffb0e0003fd%2Ftv8j9sb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Modeling Differential Equations in Financial Situations**
In this example, we will model the balance of a bank account over time using a differential equation.
**Problem Statement:**
Let \( y(t) \) represent your bank account balance, in dollars, after \( t \) years. Suppose you start with $30,000 in the account. Each year the account earns 3% interest, and you withdraw $3,000 annually.
**Objective:**
Write the differential equation that models this situation.
**Solution:**
Start by identifying all the components of the given problem:
1. Initial balance: $30,000
2. Annual interest rate: 3% (or 0.03 as a decimal)
3. Annual withdrawal: $3,000
The balance change is affected by both the interest earned and the withdrawals. The interest contributes \( 0.03 \times y(t) \) dollars each year, and the withdrawal subtracts $3,000 each year.
The rate of change of the balance, \( \frac{dy}{dt} \), can be expressed by the differential equation:
\[ \frac{dy}{dt} = 0.03y - 3000 \]
Given that the initial balance is $30,000, we have:
\[ y(0) = 30000 \]
**Summary:**
The differential equation model for the bank account balance is:
\[ \frac{dy}{dt} = 0.03y - 3000 \]
With the initial condition:
\[ y(0) = 30000 \]
This system provides a mathematical framework for understanding how your bank account balance changes over time considering interest and withdrawals.
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