**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: $30,0002. Annual interest rate: 3% (or 0.03 as a decimal)3. Annual withdrawal: $3,000The 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.

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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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

**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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