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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 9 - Differential EquationsProblema 9.R.26a
Capitolo 9, Problema 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.

Guida verificata passo dopo passo
1
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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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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Exponential Growth & Decay

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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Solutions to Basic Differential Equations

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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Initial Value Problems
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