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 9.4.4

Chapter 9, Problem 9.4.4

What is the equilibrium solution of the equation y'(t) = 3y − 9? Is it stable or unstable?

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Identify the equilibrium solution by setting the derivative equal to zero: solve the equation \(0 = 3y - 9\) for \(y\).

Rearrange the equation to isolate \(y\): \(3y = 9\), then \(y = \frac{9}{3}\).

Determine the equilibrium solution value from the above step.

To analyze stability, consider the sign of the coefficient in the differential equation \(y'(t) = 3y - 9\) when rewritten as \(y'(t) = 3(y - 3)\).

Since the coefficient 3 is positive, conclude whether the equilibrium solution is stable or unstable based on whether perturbations grow or decay over time.

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Key Concepts

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

Equilibrium Solution of a Differential Equation

An equilibrium solution occurs when the derivative y'(t) equals zero, meaning the function y(t) does not change over time. For the equation y'(t) = 3y − 9, setting y'(t) = 0 allows us to find constant solutions where the system is at rest.

Stability of Equilibrium Solutions

Stability refers to whether solutions near the equilibrium tend to move towards it (stable) or away from it (unstable) over time. This is often determined by analyzing the sign of the derivative of the right-hand side with respect to y at the equilibrium point.

Linear First-Order Differential Equations

The given equation y'(t) = 3y − 9 is a linear first-order differential equation. Understanding its structure helps in solving it and analyzing behavior near equilibrium points by using methods like separation of variables or integrating factors.

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5–10. First-order linear equations Find the general solution of the following equations.

y'(x) = −y + 2

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21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.

P′(t) = 0.05P(1−P/800); P(0) = 100, P(0) = 400, P(0) = 700

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Textbook Question

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

u''(x) = 55x⁹ + 36x⁷ - 21x⁵ + 10x⁻³

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Textbook Question

17–20. Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it.

y(t) = 8t⁶ - 3; ty'(t) - 6y(t) = 18, y(1) = 5

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Textbook Question

20–22. {Use of Tech} Solving the Gompertz equation Solve the Gompertz equation in Exercise 19 with the given values of r, K, and M₀. Then graph the solution to be sure that M(0) and lim(t→∞) M(t) are correct.

r = 0.05, K = 1200, M₀ = 90

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Textbook Question

Consider the differential equation y'(t) = t² - 3y² and the solution curve that passes through the point (3, 1). What is the slope of the curve at (3, 1)?

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