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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.31c
Capitolo 9, Problema 9.R.31c

A predator-prey model Consider the predator-prey model
x′(t) = −4x + 2xy, y′(t) = 5y − xy
c. Find the equilibrium points for the system.

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1
Write down the system of differential equations given: \(x'(t) = -4x + 2xy\) and $y'(t) = 5y - xy$.
To find the equilibrium points, set both derivatives equal to zero: \(-4x + 2xy = 0\) and \$5y - xy = 0$.
From the first equation \(-4x + 2xy = 0\), factor out \(x\): \(x(-4 + 2y) = 0\). This implies either \(x = 0\) or \(-4 + 2y = 0\).
From the second equation \$5y - xy = 0$, factor out $y\(: \)y(5 - x) = 0$. This implies either $y = 0$ or \(5 - x = 0\).
Solve the resulting cases by combining the possibilities from both equations to find all pairs \((x, y)\) that satisfy both conditions simultaneously.

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Equilibrium Points in Dynamical Systems

Equilibrium points are values where the system's variables do not change over time, meaning all derivatives are zero. For a system of differential equations, these points satisfy the condition that each derivative equals zero simultaneously. Finding equilibria helps understand the system's long-term behavior.
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Critical Points Example 2

Solving Systems of Nonlinear Equations

To find equilibrium points in nonlinear systems, set each differential equation equal to zero and solve the resulting algebraic equations simultaneously. This often involves factoring or substitution to find all possible solutions for the variables.
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Solving Logarithmic Equations

Predator-Prey Model Dynamics

The predator-prey model describes interactions between two species, where one is the predator and the other the prey. The system's equations reflect growth and decline rates influenced by their interaction terms, which are key to setting up and solving for equilibria.
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Exponential Growth & Decay
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Domanda del libro di testo

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.

a. Sketch a solution on the direction field with the initial condition y(0)=1.

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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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Domanda del libro di testo

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

b. Use Euler’s method with Δt=0.05 to compute approximations to y(0.1) and y(0.2). 

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Domanda del libro di testo

Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.

d. Complete the following sentence. The solution of the differential equation with the initial condition y(0)=A, where A is a real number, approaches the line _____ as t→∞.

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Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.

b. Use the direction field to sketch the solution curve that passes through the point (0,−1/2).

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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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