Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.1.54c

Chapter 9, Problem 9.1.54c

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v'(t) = g - bv, where v(t) is the velocity of the object for t ≥ 0, g = 9.8 m/s² is the acceleration due to gravity, and b > 0 is a constant that involves the mass of the object and the air resistance.

c. Using the graph in part (b), estimate the terminal velocity lim(t→∞) v(t).

Verified step by step guidance

Understand that the terminal velocity is the velocity value that the object approaches as time $t$ goes to infinity, meaning $\lim_{t \to \infty} v(t)$.

Recall the given differential equation: $v'(t) = g - b v(t)$, where $g$ and $b$ are constants, and $v(t)$ is the velocity at time $t$.

At terminal velocity, the velocity no longer changes, so the derivative $v'(t)$ becomes zero. Set $v'(t) = 0$ to find the equilibrium velocity: \$0 = g - b v_{terminal}$.

Solve the equation for $v_{terminal}$: $b v_{terminal} = g$, which gives $v_{terminal} = \frac{g}{b}$.

Use the graph from part (b) to estimate the value that $v(t)$ approaches as $t$ becomes very large, and compare it to the theoretical terminal velocity $\frac{g}{b}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Linear Differential Equations

This type of differential equation has the form v'(t) + p(t)v(t) = q(t). In the given model, v'(t) = g - bv can be rewritten as v'(t) + bv = g, which is linear. Understanding how to identify and solve such equations is essential for analyzing velocity over time.

Terminal Velocity

Terminal velocity is the steady-state velocity an object reaches when the net acceleration becomes zero. Mathematically, it is the limit of v(t) as t approaches infinity, where the forces of gravity and air resistance balance out, resulting in constant velocity.

Limit of a Function as t Approaches Infinity

Evaluating lim(t→∞) v(t) involves understanding how the solution to the differential equation behaves over a long time. This concept helps determine the long-term behavior of velocity, such as reaching terminal velocity in free fall with air resistance.

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Related Practice

Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

c. Repeating parts (a) and (b) using half the time step used in those calculations, again find an approximation to y(T).

y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ

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Textbook Question

{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation

dP/dt=kP(1−P/A),P0=P_0,

where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.

c. For a fixed value of K and A, describe the long-term behavior of the solutions, for any P0 with 0<P0<A.

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Textbook Question

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.

c. Find the equilibrium points for the system.

x′(t) = −3x + 6xy, y′(t) = y − 4xy

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Textbook Question

{Use of Tech} Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M(t) be the mass of a tumor at time t≥0. The relevant initial value problem is

dM/dt=−rM ln(M/K),M(0)=M0,

where r and K are positive constants and 0<M0<K.

b. Solve the initial value problem and graph the solution for r=1,K=4, and M0=1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor?

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Textbook Question

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

b. Solve the initial value problem.

A one-million-liter pond is contaminated by a chemical pollutant with a concentration of 20 g/L. The source of the pollutant is removed, and pure water is allowed to flow into the pond at a rate of 1200 L/hr. Assuming the pond is thoroughly mixed and drained at a rate of 1200 L/hr, how long does it take to reduce the concentration of the solution in the pond to 10% of the initial value?

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The general solution of the equation yy'(x) = xe⁻ʸ can be found using integration by parts.

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