Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.4.32c

Chapter 9, Problem 9.4.32c

{Use of Tech} Fish harvesting A fish hatchery has 500 fish at t=0, when harvesting begins at a rate of b>0fish/year The fish population is modeled by the initial value problem y′(t)=0.01y−b,y(0)=500 where t is measured in years.

c. Graph the solution in the case that b=60fish/year. Describe the solution.

Verified step by step guidance

Start with the given initial value problem (IVP): \(y'(t) = 0.01y - b\), with \(y(0) = 500\), and here \(b = 60\) fish/year.

Rewrite the differential equation by substituting \(b = 60\): \(y'(t) = 0.01y - 60\).

Recognize that this is a linear first-order differential equation. To solve it, first find the integrating factor \(\mu(t) = e^{-0.01t}\), which comes from the coefficient of \(y\).

Multiply both sides of the differential equation by the integrating factor to write it as a derivative of a product: \(\frac{d}{dt} \left(e^{-0.01t} y \right) = -60 e^{-0.01t}\).

Integrate both sides with respect to \(t\), then solve for \(y(t)\) using the initial condition \(y(0) = 500\). This will give the explicit solution to graph and analyze the behavior of the fish population over time.

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Key Concepts

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

Solving First-Order Linear Differential Equations

This problem involves a first-order linear differential equation of the form y' = ay + c. Understanding how to solve such equations using integrating factors or separation of variables is essential to find the explicit solution y(t), which describes the fish population over time.

Equilibrium Solutions and Stability

An equilibrium solution occurs when the population does not change over time (y' = 0). Identifying this steady state helps describe long-term behavior, such as whether the fish population stabilizes, grows, or declines, especially under constant harvesting.

Graphical Interpretation of Solutions

Graphing the solution y(t) for a specific harvesting rate (b=60) illustrates how the population changes over time. Interpreting the graph helps describe trends like population decline or approach to equilibrium, providing insight into the sustainability of harvesting.

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Related Practice

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.

(Use of Tech) Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y'(t) = -kyⁿ(t), where y(t) is the concentration of the compound, for t ≥ 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume the initial concentration of the compound is y(0) = y₀ > 0.

c. Let y₀ = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions.

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Textbook Question

Another second-order equation Consider the differential equation y''(t) + k²y(t) = 0, where k is a positive real number.

c. Give the general solution of the equation for arbitrary k > 0 and verify your conjecture.

101

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Textbook Question

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

c. y′(t) + y = √y

106

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Textbook Question

[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.

c. Graph the solutions in part (b) and describe their behavior as t increases.

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Textbook Question

{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation

m · v'(t) = mg + f(v)

mass | acceleration | external forces

where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.

c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a.

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Textbook Question

Direction field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.

c. Draw a representative direction field in the case that a<0. Show that if A>−b/a, then the solution decreases for t≥0, and that if A<−b/a, then the solution increases for t≥0.

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