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.
Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 9, Problem 9.R.28b
Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.
b. Use the solution of the logistic equation and the 2050 projected population to determine the carrying capacity.
Verified step by step guidance1
Recall the general form of the logistic growth model: \[ P(t) = \frac{K}{1 + Ae^{-rt}} \] where \(P(t)\) is the population at time \(t\), \(K\) is the carrying capacity, \(r\) is the growth rate, and \(A\) is a constant related to initial conditions.
Use the initial condition at \(t=0\) (year 1960) where \(P(0) = 435\) million to express \(A\) in terms of \(K\): \[ 435 = \frac{K}{1 + A} \implies A = \frac{K}{435} - 1 \]
Use the population at \(t=5\) (year 1965), \(P(5) = 487\) million, and substitute \(A\( into the logistic equation: \[ 487 = \frac{K}{1 + \left(\frac{K}{435} - 1\right) e^{-5r}} \] This equation relates \)K\) and \(r\).
Use the projected population for 2050, which corresponds to \(t=90\) (since 2050 - 1960 = 90), where \(P(90) = 1570\) million (1.57 billion). Substitute into the logistic model: \[ 1570 = \frac{K}{1 + \left(\frac{K}{435} - 1\right) e^{-90r}} \]
Now you have two equations involving \(K\) and \(r\) from steps 3 and 4. Solve this system of equations simultaneously to find the carrying capacity \(K\).
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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 it approaches a maximum limit called the carrying capacity. It is represented by a differential equation where growth rate decreases as population nears this limit, modeling realistic constraints like resources.
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Carrying Capacity
Carrying capacity is the maximum population size that an environment can sustain indefinitely given available resources. In the logistic model, it is the horizontal asymptote that the population approaches over time, representing the stable equilibrium point.
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Using Logistic Equation Solutions to Estimate Parameters
Solving the logistic differential equation yields a formula for population over time involving parameters like growth rate and carrying capacity. By substituting known population values at specific times, one can solve for unknown parameters such as the carrying capacity.
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Related Practice
Textbook Question
Logistic growth parameters A cell culture has a population of 20 when a nutrient solution is added at t=0. After 20 hours, the cell population is 80 and the carrying capacity of the culture is estimated to be 1600 cells.
c. After how many hours does the population reach half of the carrying capacity
Textbook Question
2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.
y′(t) + 2y = 6
Textbook Question
22–25. Equilibrium solutions Find the equilibrium solutions of the following equations and determine whether each solution is stable or unstable.
y′(t) = y(3+y)(y-5)
Textbook Question
2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.
y′(t) = √(y/t)
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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