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 9.5.30b

Chapter 9, Problem 9.5.30b

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.
b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.

x′(t) = 2x − xy, y′(t) = −y + xy

Verified step by step guidance

Identify the given system of differential equations: $x'(t) = 2x - xy$ and $y'(t) = -y + xy$.

To find the lines where $x'(t) = 0$, set the right-hand side of the first equation equal to zero: $2x - xy = 0$.

Factor the expression $2x - xy = x(2 - y) = 0$, which implies either $x = 0$ or $y = 2$. These are the lines where $x'(t) = 0$.

To find the lines where $y'(t) = 0$, set the right-hand side of the second equation equal to zero: $-y + xy = 0$.

Factor the expression $-y + xy = y(-1 + x) = 0$, which implies either $y = 0$ or $x = 1$. These are the lines where $y'(t) = 0$.

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Key Concepts

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

Predator-Prey Model

A predator-prey model describes the interaction between two species: one as prey (x) and the other as predator (y). The populations change over time according to differential equations that capture growth, death, and interaction rates. Understanding these models helps analyze population dynamics and stability.

Nullclines (Lines where derivatives are zero)

Nullclines are curves or lines in the phase plane where the rate of change of one variable is zero (e.g., x'(t) = 0 or y'(t) = 0). They help identify equilibrium points and understand system behavior by showing where populations neither increase nor decrease.

Solving for Equilibrium Conditions

To find nullclines, set each differential equation equal to zero and solve for the variables. This process reveals lines or points where population growth rates stop changing, which are critical for analyzing system stability and long-term behavior.

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Solving Separable Differential Equations

Related Practice

Textbook Question

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.)

y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1

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Textbook Question

Euler’s method on more general grids Suppose the solution of the initial value problem y'(t)=f(t,y),y(a)=A is to be approximated on the interval [a, b].

b. Write the first step of Euler’s method to compute u1.

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Textbook Question

A bad loan Consider a loan repayment plan described by the initial value problem

B'(t)=0.03B−600,B(0)=40,000,

where the amount borrowed is B(0)=\$40,000, the monthly payments are \$600, and B(t) is the unpaid balance in the loan.

b. What is the most that you can borrow under the terms of this loan without going further into debt each month?

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Textbook Question

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.

y′(t) = 2t + 1, y(0) = 0; y(t) = t² + t

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Textbook Question

brOrthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection. A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. Use the following steps to find the orthogonal trajectories of the family of ellipses 2x² + y² = a²

b. The family of trajectories orthogonal to 2x² + y² = a² satisfies the differential equation dy/dx = y/(2x). Why?

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Textbook Question

{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm.

b. Find the solution in k=0.1the case that and H=0.5m.

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