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 9.2.40a

Chapter 9, Problem 9.2.40a

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(2 - y)

Verified step by step guidance

Identify the given autonomous differential equation: \(y'(t) = y(2 - y)\).

Recall that equilibrium solutions occur where the derivative \(y'(t)\) is zero, meaning the function \(y(t)\) does not change over time.

Set the right-hand side of the differential equation equal to zero to find equilibrium points: \(y(2 - y) = 0\).

Solve the equation \(y(2 - y) = 0\) by factoring or using the zero product property, which gives the values of \(y\) where the derivative is zero.

The solutions to this equation are the equilibrium solutions \(y = 0\) and \(y = 2\), representing constant solutions where the slope field is horizontal.

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Key Concepts

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

Autonomous Differential Equations

An autonomous differential equation is one where the derivative y' depends only on the variable y, not explicitly on the independent variable t. This means the rate of change of y depends solely on y itself, simplifying analysis and allowing the direction field to be independent of t.

Equilibrium Solutions

Equilibrium solutions occur when y'(t) = 0, meaning the function y(t) remains constant over time. For autonomous equations y' = f(y), equilibrium solutions are found by solving f(y) = 0. These solutions correspond to horizontal lines in the direction field.

Stability of Equilibrium Solutions

Stability refers to whether solutions near an equilibrium tend to move towards or away from it over time. By analyzing the sign of f(y) around equilibrium points, one can determine if the equilibrium is stable (attracting), unstable (repelling), or semi-stable.

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Related Practice

Textbook Question

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

a. The differential equation y′+2y=t is first-order, linear, and separable.

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Textbook Question

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

a. Find the approximations to y(0.2) and y(0.4) using Euler’s method with time steps of Δt = 0.2, 0.1, 0.05, and 0.025.

y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²

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Textbook Question

{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation

dP/dt=kP(1−P/A),P0=P_0,

where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.

a. Find the solution of the initial value problem, for t≥0, in terms of K, A, and P0.

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Textbook Question

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

d. The direction field for the differential equation y′(t)=t+y(t) is plotted in the ty-plane.

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Textbook Question

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.

a. Identify which equation corresponds to the predator and which corresponds to the prey.

x′(t) = −3x + xy, y′(t) = 2y − xy

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Textbook Question

Stirred tank reaction A 100-L tank is filled with pure water when an inflow pipe is opened and a sugar solution with a concentration of 20 gm/L flows into the tank at a rate of 0.5 L/min. The solution is thoroughly mixed and flows out of the tank at a rate of 0.5 L/min.

c. At what time does the mass of sugar reach 95% of its steady-state level?

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