23–26. Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t≥0 graph the solution, and determine the first month in which the loan balance is zero.


B′(t) = 0.004B − 800, B(0) = 40,000

Verified step by step guidance

1. Identify the type of differential equation given: \( B'(t) = 0.004B - 800 \). This is a first-order linear ordinary differential equation.
2. Rewrite the equation in standard linear form: \( B'(t) - 0.004B = -800 \). This will help us apply the integrating factor method.
3. Calculate the integrating factor \( \mu(t) \) using the formula \( \mu(t) = e^{\int -0.004 \, dt} = e^{-0.004t} \).
4. Multiply both sides of the differential equation by the integrating factor \( e^{-0.004t} \) to get \( e^{-0.004t} B'(t) - 0.004 e^{-0.004t} B = -800 e^{-0.004t} \). Recognize that the left side is the derivative of \( e^{-0.004t} B(t) \).
5. Integrate both sides with respect to \( t \) to find \( e^{-0.004t} B(t) \), then solve for \( B(t) \). Use the initial condition \( B(0) = 40,000 \) to find the constant of integration. Finally, analyze \( B(t) \) to determine when the loan balance reaches zero by solving \( B(t) = 0 \) for \( t \).

Key Concepts

First-Order Linear Differential Equations

This type of differential equation has the form B'(t) + p(t)B = q(t). Solving it involves finding an integrating factor to simplify the equation and then integrating both sides. It models processes where the rate of change depends linearly on the current value, such as loan balances with interest and payments.

Initial Value Problems (IVP)

An IVP specifies the value of the unknown function at a particular point, here B(0) = 40,000. This condition allows us to find the unique solution to the differential equation that fits the initial state, ensuring the solution models the specific loan scenario accurately.

Interpreting and Graphing Solutions

After solving the differential equation, graphing the solution B(t) helps visualize how the loan balance changes over time. Determining when B(t) = 0 identifies the payoff time, which is crucial for understanding the loan's duration and financial planning.