Textbook Question
Growth rate functions
a. Show that the logistic growth rate function f(P)=rP(1−P/K) has a maximum value of rK/4 at the point P=K/2.
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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 9.2.18a
17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.
a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).
y'(t) = (y−2)(y+1)
Verified step by step guidance1
Identify the equilibrium solutions by setting the derivative equal to zero: \(y'(t) = 0\).
Set the right-hand side of the differential equation equal to zero: \((y - 2)(y + 1) = 0\).
Solve the equation \((y - 2)(y + 1) = 0\) for \(y\) to find the constant solutions.
The solutions to this equation give the equilibrium solutions where \(y(t)\) does not change over time.
Write the equilibrium solutions explicitly as \(y(t) = 2\) and \(y(t) = -1\) for all \(t \geq 0\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Equilibrium Solutions of Differential Equations
Equilibrium solutions occur when the derivative y'(t) equals zero for all t, meaning the solution y(t) remains constant over time. To find these, set the right-hand side of the differential equation to zero and solve for y. These solutions represent steady states where the system does not change.
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Solutions to Basic Differential Equations
Factoring and Solving Polynomial Equations
To find equilibrium points in y'(t) = (y−2)(y+1), factor the expression and set each factor equal to zero. This yields values of y that make the derivative zero. Understanding how to factor and solve polynomial equations is essential for identifying these constant solutions.
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5:02Solving Logarithmic Equations
Interpretation of Differential Equations Without Direction Fields
Even without a direction field, analyzing the sign of y'(t) around equilibrium points helps determine whether solutions increase or decrease. This qualitative analysis aids in understanding the behavior of solutions near equilibrium without graphing, focusing on algebraic manipulation and sign analysis.
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05:20Determining Differentiability Without A Graph
Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The general solution of the differential equation y'(t) = 1 is y(t) = t
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Textbook Question
[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.
a. Find the general solution of the equation and express it explicitly as a function of t in two cases: y > 0 and y < 0.
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Textbook Question
{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm.
a. Find the solution of the initial value problem.
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Textbook Question
33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].
y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ
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Textbook Question
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) = 2y + 4
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