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 9.2.44c

Chapter 9, Problem 9.2.44c

Direction field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.
c. Draw a representative direction field in the case that a<0. Show that if A>−b/a, then the solution decreases for t≥0, and that if A<−b/a, then the solution increases for t≥0.

Verified step by step guidance

Recall the given differential equation: \(y'(t) = a y + b\), with initial condition \(y(0) = A\), where \(a, b, A \in \mathbb{R}\) and \(a < 0\).

Identify the equilibrium solution by setting \(y'(t) = 0\), which gives \(a y + b = 0\). Solve for \(y\) to find the equilibrium point: \(y = -\frac{b}{a}\).

Analyze the behavior of the solution relative to the equilibrium \(y = -\frac{b}{a}\). For \(y > -\frac{b}{a}\), substitute a value greater than the equilibrium into \(y'(t) = a y + b\) and use the fact that \(a < 0\) to determine the sign of \(y'(t)\), which indicates whether the solution is increasing or decreasing.

Similarly, for \(y < -\frac{b}{a}\), substitute a value less than the equilibrium into \(y'(t)\) and analyze the sign of the derivative to determine if the solution is increasing or decreasing.

Use this information to sketch a representative direction field: draw small line segments with slopes given by \(y'(t) = a y + b\) at various points \((t, y)\), showing that solutions above the equilibrium decrease toward it, and solutions below increase toward it, confirming the stability of the equilibrium and the behavior of solutions depending on whether \(A > -\frac{b}{a}\) or \(A < -\frac{b}{a}\).

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Key Concepts

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

Direction Fields (Slope Fields)

A direction field is a graphical tool that represents the slopes of solutions to a differential equation at various points. For y' = ay + b, each point (t, y) has a slope given by ay + b. Plotting small line segments with these slopes helps visualize the behavior of solutions without solving the equation explicitly.

Equilibrium Solutions and Stability

An equilibrium solution occurs where the derivative y' = 0, here at y = -b/a. This constant solution divides the behavior of other solutions. When a < 0, the equilibrium is stable, meaning solutions near it tend to move towards it over time, influencing whether solutions increase or decrease based on their initial value relative to -b/a.

Initial Value Problems and Solution Behavior

An initial value problem specifies a starting point y(0) = A. The sign of a and the position of A relative to the equilibrium -b/a determine the solution's trend. For a < 0, if A > -b/a, the solution decreases toward the equilibrium; if A < -b/a, it increases toward it, reflecting the system's dynamics over t ≥ 0.

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Initial Value Problems

Related Practice

Textbook Question

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

b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).

y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)

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Textbook Question

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

b. The general solution of the separable equation y'(t) = t/(y' + 10y⁴) can be expressed explicitly with y in terms of t.

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Textbook Question

Euler’s method on more general grids Suppose the solution of the initial value problem y'(t)=f(t,y),y(a)=A is to be approximated on the interval [a, b].

b. Write the first step of Euler’s method to compute u1.

Textbook Question

{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation

m · v'(t) = mg + f(v)

mass | acceleration | external forces

where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.

c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a.

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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.

Textbook Question

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).