{Use of Tech} Endowment model An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem B′(t)=rB−m, for t≥0, with B(0)=B0. The constant r>0 reflects the annual interest rate, m>0 is the annual rate of withdrawal, B0 is the initial balance in the account, and t is measured in years.


a. Solve the initial value problem with r=0.05, m=\$1000/year, and B0=\$15,000 Does the balance in the account increase or decrease?

Verified step by step guidance

1. Identify the given initial value problem (IVP): \(B'(t) = rB - m\) with \(B(0) = B_0\), where \(r = 0.05\), \(m = 1000\), and \(B_0 = 15000\).
2. Recognize that this is a first-order linear differential equation. Rewrite it as \(B'(t) - rB = -m\) to match the standard form \(y' + p(t)y = q(t)\).
3. Find the integrating factor (IF), which is \(\mu(t) = e^{\int -r \, dt} = e^{-rt}\).
4. Multiply both sides of the differential equation by the integrating factor to get \(e^{-rt} B'(t) - r e^{-rt} B = -m e^{-rt}\), which simplifies to \(\frac{d}{dt} \left( e^{-rt} B \right) = -m e^{-rt}\).
5. Integrate both sides with respect to \(t\) to find \(e^{-rt} B = \int -m e^{-rt} dt + C\), then solve for \(B(t)\) and apply the initial condition \(B(0) = B_0\) to find the constant \(C\). Finally, analyze the behavior of \(B(t)\) over time to determine if the balance increases or decreases.

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. In this problem, the equation models the rate of change of the account balance over time.

Initial Value Problem (IVP)

An IVP specifies a differential equation along with an initial condition, here B(0) = B0. The initial condition allows us to find the particular solution that fits the starting balance of the account, ensuring the solution is unique and applicable to the scenario.

Interpretation of the Solution in Context

After solving the differential equation, interpreting the solution involves analyzing whether the balance increases or decreases over time. This depends on the relationship between the interest earned (rB) and the withdrawal rate (m), which determines the long-term behavior of the account.