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 9.1.53a

Chapter 9, Problem 9.1.53a

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.

where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.

a. Verify by substitution that the general solution of the equation is P(t) = K/(1 + Ce⁻ʳᵗ), where C is an arbitrary constant.

Verified step by step guidance

Start with the given differential equation, which is typically the logistic growth model: \[\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right),\] where \(P(t)\) is the population at time \(t\), \(r > 0\) is the growth rate, and \(K > 0\) is the carrying capacity.

Write down the proposed general solution: \[P(t) = \frac{K}{1 + Ce^{-rt}},\] where \(C\) is an arbitrary constant.

Compute the derivative of \(P(t)\) with respect to \(t\). Use the quotient rule or rewrite \(P(t)\) as \(K(1 + Ce^{-rt})^{-1}\) and apply the chain rule to find \[\frac{dP}{dt}.\]

Substitute \(P(t)\) and \(\frac{dP}{dt}\) back into the original differential equation \[\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)\] and simplify both sides to check if they are equal.

If both sides are equal after simplification, this verifies that the proposed \(P(t)\) satisfies the differential equation, confirming it is indeed a general solution.

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Key Concepts

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

Logistic Differential Equation

The logistic differential equation models population growth with a carrying capacity, expressed as dP/dt = rP(1 - P/K). Here, P(t) is the population at time t, r is the growth rate, and K is the maximum sustainable population. Understanding this equation is essential to analyze how populations grow and stabilize over time.

Verification by Substitution

Verification by substitution involves plugging a proposed solution into the original differential equation to check if it satisfies the equation. This method confirms the correctness of the solution without solving the equation from scratch, ensuring that the given function is indeed a solution.

General Solution and Arbitrary Constants

The general solution of a differential equation includes arbitrary constants representing a family of solutions. In this case, the constant C adjusts the initial condition of the population. Recognizing the role of arbitrary constants helps in understanding how different initial populations affect the solution.

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Verifying Solutions of Differential Equations Example 4

Related Practice

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The general solution of the differential equation y'(t) = 1 is y(t) = t

Textbook Question

[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.

a. Find the general solution of the equation and express it explicitly as a function of t in two cases: y > 0 and y < 0.

Textbook Question

Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.

a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).

Textbook Question

Direction field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.

a. Explain why y=−b/a is an equilibrium solution and corresponds to a horizontal line in the direction field.

Textbook Question

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

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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