{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\u003e0 reflects the annual interest rate, m\u003e0 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?

Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.4.34a

Chapter 9, Problem 9.4.34a

{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

Identify the given initial value problem (IVP): $B'(t) = rB - m$ with $B(0) = B_0$, where $r = 0.05$, $m = 1000$, and $B_0 = 15000$.

Recognize that this is a first-order linear differential equation. Rewrite it as $B'(t) - rB = -m$ to match the standard form $y' + p(t)y = q(t)$.

Find the integrating factor (IF), which is $\mu(t) = e^{\int -r \, dt} = e^{-rt}$.

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 $\frac{d}{dt} \left( e^{-rt} B \right) = -m e^{-rt}$.

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

a. Find the equilibrium solutions.

y′(t) = y(y - 3)

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Textbook Question

Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.

a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).

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Textbook Question

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

a. Find the general solution of the equation.

y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1

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Textbook Question

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

a. Write an initial value problem for the mass of the substance.

A one-million-liter pond is contaminated by a chemical pollutant with a concentration of 20 g/L. The source of the pollutant is removed, and pure water is allowed to flow into the pond at a rate of 1200 L/hr. Assuming the pond is thoroughly mixed and drained at a rate of 1200 L/hr, how long does it take to reduce the concentration of the solution in the pond to 10% of the initial value?

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Textbook Question

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.

a. Verify by substitution that the general solution of the equation is P(t) = K/(1 + Ce⁻ʳᵗ), where C is an arbitrary constant.

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Textbook Question

Solving Bernoulli equations Use the method outlined in Exercise 43 to solve the following Bernoulli equations.

a. y′(t) + y = 2y²

125

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