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

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.

c. Sketch the solution curve that corresponds to the initial condition y0=1.

y′(t) = 6 - 2y

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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.

d. Find lim(t→∞) P(t) and check that the result is consistent with the graph in part (c).

Textbook Question

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

c. y′(t) + y = √y

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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.

c. Why is the condition A < T₀/2 needed?

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Textbook Question

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

c. Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing?

y'(t) = cos y for |y| ≤ π

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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.