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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.4.44a
Chapter 9, Problem 9.4.44a

Solving Bernoulli equations Use the method outlined in Exercise 43 to solve the following Bernoulli equations.


a. y′(t) + y = 2y²

Verified step by step guidance
1
Identify the Bernoulli equation in the form \(y'(t) + P(t)y = Q(t)y^n\). Here, the equation is \(y'(t) + y = 2y^2\), so \(P(t) = 1\), \(Q(t) = 2\), and \(n = 2\).
Divide the entire equation by \(y^n = y^2\) (assuming \(y \neq 0\)) to rewrite it as \(y'(t) y^{-2} + y y^{-2} = 2\), which simplifies to \(y'(t) y^{-2} + y^{-1} = 2\).
Make the substitution \(z = y^{1-n} = y^{1-2} = y^{-1}\). Then, compute \(z'(t)\) in terms of \(y'(t)\): since \(z = y^{-1}\), we have \(z' = -y^{-2} y'\).
Rewrite the original equation in terms of \(z\) and \(z'\). Using \(z' = -y^{-2} y'\), rearrange to express \(y'(t) y^{-2} = -z'\). Substitute into the equation from step 2 to get \(-z' + z = 2\).
Rearrange the equation to the linear form \(z' - z = -2\). Solve this first-order linear differential equation for \(z(t)\) using an integrating factor, then substitute back \(y = z^{-1}\) to find the solution for \(y(t)\).

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Key Concepts

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

Bernoulli Differential Equation

A Bernoulli differential equation is a first-order nonlinear ODE of the form y' + P(x)y = Q(x)y^n, where n ≠ 0 or 1. It can be transformed into a linear differential equation by an appropriate substitution, making it easier to solve.
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Substitution Method for Bernoulli Equations

To solve a Bernoulli equation, use the substitution v = y^(1-n), which converts the nonlinear equation into a linear one in terms of v. This allows the use of standard methods for linear ODEs to find the solution.
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Euler's Method

Solving Linear First-Order Differential Equations

Once transformed, the equation becomes linear and can be solved using an integrating factor. The integrating factor is typically e^(∫P(x)dx), which simplifies the equation to an exact derivative, enabling integration and solution.
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Related Practice
Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

a. Find the equilibrium solutions. 


y′(t) = y(y - 3)(y + 2)

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Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

a. Find the equilibrium solutions. 


y′(t) = y(y - 3)

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Textbook Question

{Use of Tech} Endowment model An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem B′(t)=rB−m, for t≥0, with B(0)=B0. The constant r>0 reflects the annual interest rate, m>0 is the annual rate of withdrawal, B0 is the initial balance in the account, and t is measured in years.


a. Solve the initial value problem with r=0.05, m=\(1000/year, and B0=\)15,000 Does the balance in the account increase or decrease?

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Textbook Question

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

a. Write an initial value problem for the mass of the substance.


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?

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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.

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Textbook Question

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.

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