Skip to main content
Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 9 - Differential Equations문제 9.4.44c
9장, 문제 9.4.44c

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


c. y′(t) + y = √y

검증된 단계별 안내
1
Identify the given differential equation: \(y'(t) + y = \sqrt{y}\). Notice that this is a Bernoulli equation because it can be written in the form \(y' + P(t)y = Q(t)y^n\) with \(n = \frac{1}{2}\).
Rewrite the equation explicitly as \(y' + y = y^{1/2}\). Here, \(P(t) = 1\), \(Q(t) = 1\), and \(n = \frac{1}{2}\).
Make the substitution \(z = y^{1 - n} = y^{1 - \frac{1}{2}} = y^{\frac{1}{2}}\). This substitution transforms the nonlinear equation into a linear one in terms of \(z\).
Differentiate \(z\) with respect to \(t\): \(\frac{dz}{dt} = \frac{1}{2} y^{-\frac{1}{2}} y'\). Solve this expression for \(y'\) to substitute back into the original equation.
Substitute \(y'\) in the original equation with the expression in terms of \(z\) and \(\frac{dz}{dt}\), then simplify to get a linear differential equation in \(z\). Solve this linear equation using an integrating factor.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Bernoulli Differential Equation

A Bernoulli differential equation is a nonlinear first-order ODE of the form y' + P(x)y = Q(x)y^n, where n ≠ 0 or 1. It can be transformed into a linear equation by an appropriate substitution, making it easier to solve.
추천 영상:
07:39
Classifying Differential Equations

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 differential equation in terms of v. This allows the use of standard methods for linear ODEs.
추천 영상:
07:33
Euler's Method

Solving Linear First-Order Differential Equations

Once transformed, the equation becomes linear and can be solved using an integrating factor, μ(t) = e^(∫P(t)dt). Multiplying through by μ(t) simplifies the equation, enabling integration and solution for v(t), and then y(t).
추천 영상:
06:06
Solving Separable Differential Equations
관련 실천
교과서 질문

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.

c. Sketch the solution curve that corresponds to the initial condition y0=1. 


y′(t) = 6 - 2y

55
views
교과서 질문

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) Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y'(t) = -kyⁿ(t), where y(t) is the concentration of the compound, for t ≥ 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume the initial concentration of the compound is y(0) = y₀ > 0.


c. Let y₀ = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions. 

48
views
교과서 질문

Another second-order equation Consider the differential equation y''(t) + k²y(t) = 0, where k is a positive real number.

c. Give the general solution of the equation for arbitrary k > 0 and verify your conjecture.

101
views
교과서 질문

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.


c. Why is the condition A < T₀/2 needed?

49
views
교과서 질문

{Use of Tech} Fish harvesting A fish hatchery has 500 fish at t=0, when harvesting begins at a rate of b>0fish/year The fish population is modeled by the initial value problem y′(t)=0.01y−b,y(0)=500 where t is measured in years.


c. Graph the solution in the case that b=60fish/year. Describe the solution.

36
views
교과서 질문

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

c. Graph the solutions in part (b) and describe their behavior as t increases. 

43
views