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Explain how a stirred tank reaction works.
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 9, Problema 9.5.15
15–16. {Use of Tech} Solving logistic equations Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let r be the natural growth rate, K the carrying capacity, and P₀ the initial population.
r=0.2, K=300, P₀=50
Guida verificata passo dopo passo1
Start by writing the general form of the logistic differential equation: \[\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)\] where \(P(t)\) is the population at time \(t\), \(r\) is the natural growth rate, and \(K\) is the carrying capacity.
Substitute the given parameter values into the logistic equation: \[\frac{dP}{dt} = 0.2P\left(1 - \frac{P}{300}\right)\].
Set up the initial value problem (IVP) by including the initial condition: \[P(0) = 50\].
To solve the IVP, separate variables and integrate: rewrite the equation as \[\frac{dP}{P(1 - P/300)} = 0.2\,dt\] and then perform partial fraction decomposition on the left side before integrating both sides.
After integrating, solve for \(P(t)\) explicitly in terms of \(t\) and the constants, then use the initial condition \(P(0) = 50\) to find the constant of integration. This will give you the logistic growth function describing the population over time.
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Logistic Growth Model
The logistic growth model describes population growth that starts exponentially but slows as the population approaches a maximum limit called the carrying capacity (K). It is represented by the differential equation dP/dt = rP(1 - P/K), where r is the natural growth rate and P is the population at time t.
Video consigliato:
09:29
Exponential Growth & Decay
Solving Initial Value Problems (IVPs)
An initial value problem involves solving a differential equation with a given initial condition, such as P(0) = P₀. Solving the logistic equation IVP means finding a function P(t) that satisfies both the logistic differential equation and the initial population value.
Video consigliato:
Percorso guidato
05:03Initial Value Problems
Graphing Solutions of Differential Equations
Graphing the solution to a logistic equation shows how the population changes over time, typically starting at P₀, growing rapidly, and leveling off near the carrying capacity K. This visual helps interpret the behavior and stability of the population model.
Video consigliato:
04:00Solutions to Basic Differential Equations
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Solution of the logistic equation Use separation of variables to show that the solution of the initial value problem
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