Textbook Question
11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.
y′(x) = x/y, y(2) = 4
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 9, Problem 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.
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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Related Practice
Textbook Question
2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.
y′(t) = (2t+1)(y²+1)
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Textbook Question
A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ
a. Show that the left side of the equation can be written as the derivative of a single term.
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Textbook Question
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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Textbook Question
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
Textbook Question
A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ
c. Find the solution that satisfies the condition y(1) = 0