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 9.1.56b

Chapter 9, Problem 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.

Verified step by step guidance

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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Key Concepts

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

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.

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.

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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Related Practice

Textbook Question

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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Textbook Question

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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Textbook Question

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.

Textbook Question

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 one-million-liter pond is contaminated by a chemical pollutant with a concentration of 20 g/L. The source of the pollutant is removed, and pure water is allowed to flow into the pond at a rate of 1200 L/hr. Assuming the pond is thoroughly mixed and drained at a rate of 1200 L/hr, how long does it take to reduce the concentration of the solution in the pond to 10% of the initial value?

Textbook Question

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(y+3)(4-y)

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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.