Domanda del libro di testo
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample
b. If k>0 and b>0 then y(t)=0 is never a solution of y'(t)=ky−b.
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 9, Problema 9.1.56b
52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.
{Use of Tech} Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t ≥ 0. The relevant initial value problem is:
dM/dt = -rM(t)ln(M(t)/K), M(0) = M₀,
where r and K are positive constants and 0 < M₀ < K.
b. Graph the solution for M₀ = 100 and r = 0.05.
Guida verificata passo dopo passo1
Identify the given differential equation: \(\frac{dM}{dt} = -r M(t) \ln\left(\frac{M(t)}{K}\right)\) with initial condition \(M(0) = M_0\).
Recognize that this is a Gompertz growth model, a type of nonlinear differential equation often solved by separating variables or using an integrating factor after substitution.
Rewrite the equation by introducing a substitution to simplify the logarithmic term, for example, let \(y(t) = \ln\left(\frac{M(t)}{K}\right)\), then express \(\frac{dM}{dt}\) in terms of \(y\) and \(\frac{dy}{dt}\).
Solve the resulting differential equation for \(y(t)\), then back-substitute to find \(M(t)\) in terms of \(t\), \(r\), \(K\), and \(M_0\).
To graph the solution for \(M_0 = 100\) and \(r = 0.05\), choose a value for \(K\) (if not given), compute \(M(t)\) over a range of \(t\) values, and plot \(M(t)\) versus \(t\) to visualize tumor growth over time.
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Differential Equations and Initial Value Problems
A differential equation relates a function with its derivatives, describing how the function changes over time. An initial value problem specifies the value of the function at a starting point, allowing for a unique solution. Solving such problems involves finding a function that satisfies both the differential equation and the initial condition.
Video consigliato:
Percorso guidato
05:03
Initial Value Problems
Gompertz Growth Model
The Gompertz growth model describes growth processes that slow down as they approach a limiting value, often used in biology for tumor growth. It is characterized by a differential equation where the growth rate decreases logarithmically as the mass approaches a carrying capacity K. This model captures realistic saturation effects in growth.
Video consigliato:
09:29Exponential Growth & Decay
Graphing Solutions of Differential Equations
Graphing the solution of a differential equation involves plotting the function that satisfies the equation over a range of values. This visualizes how the quantity evolves over time, helping to interpret behavior such as growth, decay, or stabilization. Numerical methods or software tools are often used when explicit solutions are complex.
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04:00Solutions to Basic Differential Equations
Pratica correlata
Domanda del libro di testo
33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).
y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)
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Domanda del libro di testo
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The general solution of the separable equation y'(t) = t/(y' + 10y⁴) can be expressed explicitly with y in terms of t.
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Domanda del libro di testo
23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.
b. Solve the initial value problem.
A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.
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Domanda del libro di testo
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 − 4xy, y′(t) = −y + 2xy
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Domanda del libro di testo
17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.
b. In what regions are solutions increasing? Decreasing?
y'(t) = (y−1)(1+y)
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