Skip to main content
Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.2.44a
Chapter 9, Problem 9.2.44a

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.
a. Explain why y=−b/a is an equilibrium solution and corresponds to a horizontal line in the direction field.

Verified step by step guidance
1
Recall that an equilibrium solution to a differential equation is a constant solution where the derivative is zero for all values of \( t \). This means \( y'(t) = 0 \) along that solution.
Given the differential equation \( y'(t) = a y + b \), set \( y'(t) = 0 \) to find the equilibrium solution: \( 0 = a y + b \).
Solve for \( y \) to get \( y = -\frac{b}{a} \). This value of \( y \) makes the derivative zero, so the solution is constant and does not change with \( t \).
Since \( y = -\frac{b}{a} \) is constant, its graph is a horizontal line in the \( t y \)-plane, which corresponds to a horizontal line in the direction field.
In the direction field, the slope at every point on this line is zero, confirming that \( y = -\frac{b}{a} \) is an equilibrium solution.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m

Key Concepts

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

Equilibrium Solution

An equilibrium solution to a differential equation is a constant solution where the derivative is zero. For y'(t) = ay + b, setting y' = 0 gives y = -b/a, meaning the solution does not change over time and remains constant.
Recommended video:
04:00
Solutions to Basic Differential Equations

Direction Field

A direction field is a graphical representation showing the slope of solutions to a differential equation at various points. At an equilibrium solution, the slope is zero, so the direction field shows horizontal line segments, indicating no change in y.
Recommended video:
05:45
Understanding Slope Fields

Initial Value Problem (IVP)

An IVP specifies a differential equation along with an initial condition, such as y(0) = A. This condition determines a unique solution curve passing through the point (0, A) in the direction field.
Recommended video:
Guided course
05:03
Initial Value Problems
Related Practice
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.

Textbook Question

Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.


a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).

Textbook Question

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

a. Find the general solution of the equation.


e⁻ʸᐟ²y'(x) = 4x sin x² − x; y(0) = 0, y(0) = ln(1/4), y(√(π/2)) = 0


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.

1
views
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.


where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.


a. Verify by substitution that the general solution of the equation is P(t) = K/(1 + Ce⁻ʳᵗ), where C is an arbitrary constant.

Textbook Question

Stirred tank reaction A 100-L tank is filled with pure water when an inflow pipe is opened and a sugar solution with a concentration of 20 gm/L flows into the tank at a rate of 0.5 L/min. The solution is thoroughly mixed and flows out of the tank at a rate of 0.5 L/min.


c. At what time does the mass of sugar reach 95% of its steady-state level?

1
views