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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 9 - Differential EquationsProblema 9.R.19d
Capitolo 9, Problema 9.R.19d

Direction fields Consider the direction field for the equation y′=y(2−y) shown in the figure and initial conditions of the form y(0)=A.
d. For what values of A are the corresponding solutions decreasing, for t≥0
Direction field graph showing slope vectors for y′=y(2−y) with axes labeled t and y from -3 to 3.

Guida verificata passo dopo passo
1
Identify the differential equation given: \(y' = y(2 - y)\). This represents the slope of the solution curves at any point \((t, y)\).
Recall that the solution is decreasing where the derivative \(y'\) is negative. So, we need to find where \(y(2 - y) < 0\).
Analyze the inequality \(y(2 - y) < 0\). This product is negative when one factor is positive and the other is negative. So, either \(y < 0\) and \(2 - y > 0\), or \(y > 0\) and \(2 - y < 0\).
Simplify the conditions: For \(y < 0\), \(2 - y\) is always positive, so \(y' < 0\) for all \(y < 0\). For \(y > 0\), \(2 - y < 0\) means \(y > 2\). So, \(y' < 0\) when \(y > 2\).
Conclude that the solutions are decreasing for \(t \geq 0\) if the initial condition \(A = y(0)\) satisfies \(A < 0\) or \(A > 2\).

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Concetti chiave

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Direction Fields and Slope Interpretation

A direction field visualizes the slopes of solutions to a differential equation at various points. Each small line segment represents the slope y' at that (t, y) coordinate, indicating the behavior of solution curves without solving the equation explicitly.
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05:45
Understanding Slope Fields

Equilibrium Solutions and Stability

Equilibrium solutions occur where y' = 0, meaning the solution is constant. For y' = y(2 - y), equilibria are at y = 0 and y = 2. Stability depends on the sign of y' near these points, determining if solutions approach or move away from equilibria.
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Solutions to Basic Differential Equations

Monotonicity of Solutions Based on Initial Conditions

The sign of y' determines whether solutions increase or decrease. For initial value y(0) = A, if y' < 0 for t ≥ 0, the solution decreases. Analyzing y' = y(2 - y) shows that solutions decrease when A > 2 or A < 0, as the product y(2 - y) is negative in these intervals.
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Percorso guidato
05:03
Initial Value Problems
Pratica correlata
Domanda del libro di testo

Euler’s metho d Consider the initial value problem y′(t)=1/2y,y(0)=1. 

a. Use Euler’s method with Δt=0.1 to compute approximations to y(0.1) and y(0.2). 

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12–16. Sketching direction fields Use the window [-2, 2] x [-2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed.

y(x) = sin y, y(−2) = 1/2

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Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.

e. Discuss some possible shortcomings of this model. Why might the carrying capacity be either greater than or less than the value predicted by the model?

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11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.

y′(t) = -3y + 9, y(0) = 4

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Domanda del libro di testo

What are the assumptions underlying the predator-prey model discussed in this section?

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7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁t⁵ + C₂t⁻⁴ - t³; t²u''(t) - 20u(t) = 14t³

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