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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.R.26a
Chapter 9, Problem 9.R.26a

Logistic growth The population of a rabbit community is governed by the initial value problem
P′(t) = 0.2 P (1 − P/1200), P(0) = 50
a. Find the equilibrium solutions.

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Identify the differential equation given: \(P\'(t) = 0.2 P \left(1 - \frac{P}{1200}\right)\).
Recall that equilibrium solutions occur when the population does not change over time, meaning \(P\'(t) = 0\).
Set the right-hand side of the differential equation equal to zero: \(0.2 P \left(1 - \frac{P}{1200}\right) = 0\).
Solve the equation \(0.2 P \left(1 - \frac{P}{1200}\right) = 0\) by setting each factor equal to zero separately: \(P = 0\) or \(1 - \frac{P}{1200} = 0\).
From \(1 - \frac{P}{1200} = 0\), solve for \(P\) to find the second equilibrium solution: \(P = 1200\).

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Key Concepts

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

Logistic Growth Model

The logistic growth model describes population growth that starts exponentially but slows as the population approaches a carrying capacity. It is represented by the differential equation P'(t) = rP(1 - P/K), where r is the growth rate and K is the carrying capacity. This model reflects limited resources affecting growth.
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Equilibrium Solutions of Differential Equations

Equilibrium solutions occur when the rate of change is zero, meaning P'(t) = 0. For population models, these solutions represent steady states where the population remains constant over time. Finding equilibria involves setting the differential equation's right side to zero and solving for P.
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Initial Value Problem (IVP)

An initial value problem specifies a differential equation along with an initial condition, such as P(0) = 50. This condition helps determine a unique solution curve from the family of possible solutions. Understanding IVPs is essential for applying and interpreting models in real-world contexts.
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Related Practice
Textbook Question

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.

c. When does the temperature of the coffee reach 50°C?

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Textbook Question

Euler’s metho d Consider the initial value problem y′(t)=1/2y,y(0)=1. 

a. Use Euler’s method with Δt=0.1 to compute approximations to y(0.1) and y(0.2). 

Textbook Question

A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ

a. Show that the left side of the equation can be written as the derivative of a single term.

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Textbook Question

Euler’s method Consider the initial value problem y′(t)=1/2y,y(0)=1. 

b. Use Euler’s method with Δt=0.05 to compute approximations to y(0.1) and y(0.2). 

Textbook Question

Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.

d. Complete the following sentence. The solution of the differential equation with the initial condition y(0)=A, where A is a real number, approaches the line _____ as t→∞.

Textbook Question

Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.

b. Use the direction field to sketch the solution curve that passes through the point (0,−1/2).

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