Textbook Question
Growth rate functions
a. Show that the logistic growth rate function f(P)=rP(1−P/K) has a maximum value of rK/4 at the point P=K/2.
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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 9.3.43a
42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.
a. Find the general solution of the equation.
e⁻ʸᐟ²y'(x) = 4x sin x² − x; y(0) = 0, y(0) = ln(1/4), y(√(π/2)) = 0
Verified step by step guidance1
Rewrite the given differential equation \(e^{-\frac{y}{2}} y'(x) = 4x \sin(x^2) - x\) in the form \(\frac{dy}{dx} = e^{\frac{y}{2}} (4x \sin(x^2) - x)\) to isolate \(y'\) on one side.
Recognize that the equation is separable, so rearrange terms to separate variables: \(e^{-\frac{y}{2}} dy = (4x \sin(x^2) - x) dx\).
Integrate both sides: \(\int e^{-\frac{y}{2}} dy = \int (4x \sin(x^2) - x) dx\). For the left side, use substitution \(u = -\frac{y}{2}\); for the right side, split the integral into two parts and use substitution for \(\int 4x \sin(x^2) dx\).
After integrating, include the constant of integration \(C\) and write the implicit general solution relating \(y\) and \(x\).
Use the given initial conditions \(y(0) = 0\), \(y(0) = \ln(\frac{1}{4})\), and \(y(\sqrt{\frac{\pi}{2}}) = 0\) to solve for the constant \(C\) and verify the solution's consistency.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Separable Differential Equations
A separable differential equation can be written as the product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently, facilitating the solution of the differential equation.
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Solving Separable Differential Equations
Implicit Solutions
An implicit solution to a differential equation is a relation involving both x and y that defines y implicitly as a function of x. Unlike explicit solutions, implicit solutions may not isolate y on one side but still satisfy the differential equation and initial conditions.
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Initial Conditions and Particular Solutions
Initial conditions specify the value of the solution at a particular point, allowing determination of the constant of integration in the general solution. Applying these conditions yields a unique particular solution that fits the given problem context.
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Related Practice
Textbook Question
43–44. Motion in a gravitational field: An object is fired vertically upward with initial velocity v(0)=v₀ from initial position s(0)=s₀.
a. For the following values of v₀ and s₀, find the position and velocity functions for all times at which the object is above the ground (s = 0).
v₀ = 49 m/s, s₀ = 60 m
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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.
a. Find the solution of the initial value problem.
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Textbook Question
29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.
a. Find the approximations to y(0.2) and y(0.4) using Euler’s method with time steps of Δt = 0.2, 0.1, 0.05, and 0.025.
y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²
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Textbook Question
{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation
dP/dt=kP(1−P/A),P0=P_0,
where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.
a. Find the solution of the initial value problem, for t≥0, in terms of K, A, and P0.
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Textbook Question
33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].
y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ
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