First Course in Differential Equations (Instructor's)
11th Edition
ISBN: 9781305965775
Author: ZILL
Publisher: CENGAGE L
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Textbook Question
Chapter 5.1, Problem 59E
Show that if L, R, E0, and γ are constant, then the amplitude of the steady-state current in Example 10 is a maximum when the capacitance is C = 1/Lγ2.
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Consider two populations that can be modeled by the logistic equation with
the same carrying capacity, but with different values of k. Which population
approaches the carrying capacity most quickly, the one with the smaller or larger
value of k? You may want to think geometrically (check out some slope fields),
analytically (look at the form of the solution), or num
into the solution).
erically (plug some t values
Biologists release two otters (a type of water animal) into a river. They believe
that the river could support a maximum population of 500 otters, and that with
unlimited resources the otter population would increase by 50% each year.
i. Write down a Pielou logistic equation to model the yearly otter population, of
the form
Yn+1
Xyn
1+ Byn
for suitable a, ẞER.
ii. By solving this equation, make a prediction of the population after 5 years.
Some populations initially grow exponentially but eventually level off. Equations of the formP (t) = M/(1 + Ae-kt)where M, A, and k are positive constants, are called logistic equations andare often used to model such populations. Here M is called the carryingcapacity and represents the maximum population size that can be supported,and A =(M−P0 )/Powhere P0 is the initial population.(a) Compute lim
t→∞P (t).
Explain why your answer is to be expected.
(b)
Compute limM→∞P (t). (Note that A is defined in terms of M.) What kindof function is your result?
Chapter 5 Solutions
First Course in Differential Equations (Instructor's)
Ch. 5.1 - 5.1.1 Spring/Mass systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A force...Ch. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - 5.1.1Spring/Mass Systems: Free Undamped Motion A...
Ch. 5.1 - A mass weighing 64 pounds stretches a spring 0.32...Ch. 5.1 - A mass of 1 slug is suspended from a spring whose...Ch. 5.1 - Prob. 13ECh. 5.1 - 5.1.1 Spring/Mass systems: Free Undamped Motion A...Ch. 5.1 - Solve Problem 13 again, but this time assume that...Ch. 5.1 - Prob. 16ECh. 5.1 - Spring/Mass Systems: Free Undamped Motion Find the...Ch. 5.1 - Prob. 18ECh. 5.1 - Spring/Mass Systems: Free Undamped Motion A model...Ch. 5.1 - 5.1.1Spring/Mass Systems: Free Undamped Motion A...Ch. 5.1 - 5.1.2 Spring/Mass systems: Free Damped Motion In...Ch. 5.1 - Spring/Mass Systems: Free Damped Motion In...Ch. 5.1 - Spring/Mass Systems: Free Damped Motion In...Ch. 5.1 - Spring/Mass Systems: Free Damped Motion In...Ch. 5.1 - Spring/Mass System: Free Damped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Damped Motion A 4-foot...Ch. 5.1 - A 1-kilogram mass is attached to a spring whose...Ch. 5.1 - A 1-kilogram mass is attached to a spring whose...Ch. 5.1 - Spring/Mass Systems: Free Damped Motion A force of...Ch. 5.1 - After a mass weighing 10 pounds is attached to a...Ch. 5.1 - Spring/Mass Systems: Free Damped Motion A mass...Ch. 5.1 - Prob. 32ECh. 5.1 - Spring/Mass Systems: Free Damped Motion A mass...Ch. 5.1 - A mass of 1 slug is attached to a spring whose...Ch. 5.1 - Spring/Mass Systems: Driven Motion A mass of 1...Ch. 5.1 - In Problem 35 determine the equation of motion if...Ch. 5.1 - Spring/Mass Systems: Driven Motion When a mass of...Ch. 5.1 - Prob. 38ECh. 5.1 - Spring/Mass Systems: Driven Motion A mass m is...Ch. 5.1 - A mass of 100 grams is attached to a spring whose...Ch. 5.1 - Prob. 41ECh. 5.1 - Prob. 42ECh. 5.1 - Series Circuit Analogue (a) Show that the solution...Ch. 5.1 - Compare the result obtained in part (b) of Problem...Ch. 5.1 - (a) Show that x(t) given in part (a) of Problem 43...Ch. 5.1 - Series Circuit Analogue Find the charge on the...Ch. 5.1 - Series Circuit Analogue Find the charge on the...Ch. 5.1 - Series Circuit Analogue In Problems 51 and 52 find...Ch. 5.1 - In Problems 51 and 52 find the charge on the...Ch. 5.1 - Series Circuit Analogue Find the steady-state...Ch. 5.1 - Prob. 54ECh. 5.1 - Prob. 55ECh. 5.1 - Prob. 56ECh. 5.1 - Find the charge on the capacitor in an LRC-series...Ch. 5.1 - Show that if L, R, C, and E0 are constant, then...Ch. 5.1 - Show that if L, R, E0, and are constant, then the...Ch. 5.1 - Series Circuit Analogue Find the charge on the...Ch. 5.1 - Prob. 61ECh. 5.1 - Prob. 62ECh. 5.2 - (a) The beam is embedded at its left end and free...Ch. 5.2 - Prob. 2ECh. 5.2 - (a) The beam is embedded at its left end and...Ch. 5.2 - (a) The beam is embedded at its left end and...Ch. 5.2 - Prob. 6ECh. 5.2 - A cantilever beam of length L is embedded at its...Ch. 5.2 - Prob. 8ECh. 5.2 - In Problems 920 find the eigenvalues and...Ch. 5.2 - In Problems 920 find the eigenvalues and...Ch. 5.2 - In Problems 920 find the eigenvalues and...Ch. 5.2 - In Problems 920 find the eigenvalues and...Ch. 5.2 - In Problems 920 find the eigenvalues and...Ch. 5.2 - Prob. 14ECh. 5.2 - Prob. 15ECh. 5.2 - Prob. 16ECh. 5.2 - In Problems 920 find the eigenvalues and...Ch. 5.2 - Eigenvalues and Eigenfunctions In Problems 920...Ch. 5.2 - Eigenvalues and Eigenfunctions In Problems 920...Ch. 5.2 - Prob. 20ECh. 5.2 - In Problems 21 and 22 find the eigenvalues and...Ch. 5.2 - In Problems 21 and 22 find the eigenvalues and...Ch. 5.2 - Prob. 23ECh. 5.2 - The critical loads of thin columns depend on the...Ch. 5.2 - Prob. 25ECh. 5.2 - Prob. 27ECh. 5.2 - Prob. 28ECh. 5.2 - Additional Boundary-Value Problems Temperature in...Ch. 5.2 - Additional Boundary-Value Problems Temperature In...Ch. 5.2 - Rotation of a Shaft Suppose the x-axis on the...Ch. 5.2 - Prob. 32ECh. 5.2 - Discussion Problems Simple Harmonic Motion The...Ch. 5.2 - Prob. 34ECh. 5.2 - Prob. 35ECh. 5.2 - Prob. 36ECh. 5.2 - Prob. 37ECh. 5.2 - Prob. 38ECh. 5.3 - Find a linearization of the differential equation...Ch. 5.3 - (a) Use the substitution v = dy/dt to solve (13)...Ch. 5.3 - Prob. 15ECh. 5.3 - A uniform chain of length L, measured in feet, is...Ch. 5.3 - Pursuit curve In a naval exercise a ship S1 is...Ch. 5.3 - Pursuit curve In another naval exercise a...Ch. 5.3 - The ballistic pendulum Historically, in order to...Ch. 5.3 - Prob. 21ECh. 5 - If a mass weighing 10 pounds stretches a spring...Ch. 5 - The period of simple harmonic motion of mass...Ch. 5 - The differential equation of a spring/mass system...Ch. 5 - Pure resonance cannot take place in the presence...Ch. 5 - Prob. 5RECh. 5 - Prob. 6RECh. 5 - Prob. 7RECh. 5 - Prob. 8RECh. 5 - Prob. 9RECh. 5 - Prob. 10RECh. 5 - A free undamped spring/mass system oscillates with...Ch. 5 - A mass weighing 12 pounds stretches a spring 2...Ch. 5 - A force of 2 pounds stretches a spring 1 foot....Ch. 5 - A mass weighing 32 pounds stretches a spring 6...Ch. 5 - A spring with constant k = 2 is suspended in a...Ch. 5 - Prob. 16RECh. 5 - A mass weighing 4 pounds stretches a spring 18...Ch. 5 - Find a particular solution for x + 2x + 2x = A,...Ch. 5 - Prob. 19RECh. 5 - Prob. 20RECh. 5 - A series circuit contains an inductance of L= 1 h,...Ch. 5 - (a) Show that the current i(t) in an LRC-series...Ch. 5 - Consider the boundary-value problem...Ch. 5 - Suppose a mass m lying on a flat dry frictionless...Ch. 5 - Prob. 26RECh. 5 - Suppose the mass m in the spring/mass system in...Ch. 5 - Prob. 28RECh. 5 - Prob. 29RECh. 5 - Spring pendulum The rotational form of Newtons...Ch. 5 - Prob. 31RECh. 5 - Galloping Gertie Bridges are good examples of...
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- Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such that the initial population at time t=0 is P(0)=P0. Show algebraically that cP(t)P(t)=cP0P0ebt .arrow_forwardIn Exercises 7-10, find a linear equation that has the same solution set as the given equation (possibly with some restrictions on the variables). 10.arrow_forwardDoes the equation y=2.294e0.654t representcontinuous growth, continuous decay, or neither?Explain.arrow_forward
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