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 9.5.30b

Chapter 9, Problem 9.5.30b

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.
b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.

x′(t) = 2x − xy, y′(t) = −y + xy

Verified step by step guidance

Identify the given system of differential equations: \(x'(t) = 2x - xy\) and \(y'(t) = -y + xy\).

To find the lines where \(x'(t) = 0\), set the right-hand side of the first equation equal to zero: \(2x - xy = 0\).

Factor the expression \(2x - xy = x(2 - y) = 0\), which implies either \(x = 0\) or \(y = 2\). These are the lines where \(x'(t) = 0\).

To find the lines where \(y'(t) = 0\), set the right-hand side of the second equation equal to zero: \(-y + xy = 0\).

Factor the expression \(-y + xy = y(-1 + x) = 0\), which implies either \(y = 0\) or \(x = 1\). These are the lines where \(y'(t) = 0\).

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Key Concepts

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

Predator-Prey Model

A predator-prey model describes the interaction between two species: one as prey (x) and the other as predator (y). The populations change over time according to differential equations that capture growth, death, and interaction rates. Understanding these models helps analyze population dynamics and stability.

Nullclines (Lines where derivatives are zero)

Nullclines are curves or lines in the phase plane where the rate of change of one variable is zero (e.g., x'(t) = 0 or y'(t) = 0). They help identify equilibrium points and understand system behavior by showing where populations neither increase nor decrease.

Solving for Equilibrium Conditions

To find nullclines, set each differential equation equal to zero and solve for the variables. This process reveals lines or points where population growth rates stop changing, which are critical for analyzing system stability and long-term behavior.

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Solving Separable Differential Equations

Related Practice

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. Euler’s method is used to compute exact values of the solution of an initial value problem.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The solution of a stirred tank initial value problem always approaches a constant as t→∞

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Textbook Question

Blowup in finite time Consider the initial value problem y'(t) = yⁿ + 1, y(0) = y₀, where n is a positive integer.

b. Solve the initial value problem with n = 2 and y₀ = 1/√2.

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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.

b. Sketch the direction field, for t≥0.

y′(t) = 2y + 4

Textbook Question

b. Sketch the direction field, for t≥0.

y′(t) = 6 - 2y

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Textbook Question

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.

y′(t) = 2t + 1, y(0) = 0; y(t) = t² + t

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