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 9.2.38a

Chapter 9, Problem 9.2.38a

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

Verified step by step guidance

Identify the given differential equation: \(y'(t) = 2y + 4\).

Recall that equilibrium solutions occur when the derivative \(y'(t)\) is zero, meaning the function \(y(t)\) does not change over time.

Set the right-hand side of the differential equation equal to zero to find equilibrium points: \(2y + 4 = 0\).

Solve the algebraic equation for \(y\) to find the equilibrium solution(s).

Interpret the solution(s) as constant functions \(y(t) = y_0\) where \(y_0\) satisfies the equation, representing horizontal lines in the direction field.

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Key Concepts

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

Autonomous Differential Equations

An autonomous differential equation is one where the derivative y' depends only on the variable y, not explicitly on the independent variable t. This means the rate of change of y is determined solely by y itself, simplifying analysis and allowing the use of phase line methods.

Equilibrium Solutions

Equilibrium solutions occur when y' = 0, meaning the function y(t) remains constant over time. For autonomous equations y' = f(y), equilibrium points are found by solving f(y) = 0. These solutions correspond to horizontal lines in the direction field where the system is at rest.

Direction Fields and Stability

Direction fields graphically represent the slope y' at various points (t, y). For autonomous equations, slopes depend only on y, producing horizontal patterns independent of t. Equilibrium solutions appear as horizontal lines, and analyzing nearby slopes helps determine their stability.

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Understanding Slope Fields

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

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)

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Textbook Question

46–48. Analyzing models The following models were discussed in Section 9.1 and reappear in later sections of this chapter. In each case, carry out the indicated analysis using direction fields.

Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m′(t)+km(t)=I, where m(t) is the mass of the drug in the blood at time t≥0, K is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. Let I=10mg/hr and k=0.05 hr^−1.

a. Draw the direction field, for 0≤t≤100, 0≤y≤600.

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Textbook Question

Consider the differential equation y'(t)+9y(t)=10.

a. How many arbitrary constants appear in the general solution of the differential equation?

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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) = 6 - 2y

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