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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 9.3.43a
9장, 문제 9.3.43a

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


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Rewrite the given differential equation \(e^{-\frac{y}{2}} y'(x) = 4x \sin(x^2) - x\) in the form \(\frac{dy}{dx} = e^{\frac{y}{2}} (4x \sin(x^2) - x)\) to isolate \(y'\) on one side.
Recognize that the equation is separable, so rearrange terms to separate variables: \(e^{-\frac{y}{2}} dy = (4x \sin(x^2) - x) dx\).
Integrate both sides: \(\int e^{-\frac{y}{2}} dy = \int (4x \sin(x^2) - x) dx\). For the left side, use substitution \(u = -\frac{y}{2}\); for the right side, split the integral into two parts and use substitution for \(\int 4x \sin(x^2) dx\).
After integrating, include the constant of integration \(C\) and write the implicit general solution relating \(y\) and \(x\).
Use the given initial conditions \(y(0) = 0\), \(y(0) = \ln(\frac{1}{4})\), and \(y(\sqrt{\frac{\pi}{2}}) = 0\) to solve for the constant \(C\) and verify the solution's consistency.

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주요 개념

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

Separable Differential Equations

A separable differential equation can be written as the product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently, facilitating the solution of the differential equation.
추천 영상:
06:06
Solving Separable Differential Equations

Implicit Solutions

An implicit solution to a differential equation is a relation involving both x and y that defines y implicitly as a function of x. Unlike explicit solutions, implicit solutions may not isolate y on one side but still satisfy the differential equation and initial conditions.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative

Initial Conditions and Particular Solutions

Initial conditions specify the value of the solution at a particular point, allowing determination of the constant of integration in the general solution. Applying these conditions yields a unique particular solution that fits the given problem context.
추천 영상:
가이드 코스
05:03
Initial Value Problems
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Growth rate functions

a. Show that the logistic growth rate function f(P)=rP(1−P/K) has a maximum value of rK/4 at the point P=K/2.

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교과서 질문

43–44. Motion in a gravitational field: An object is fired vertically upward with initial velocity v(0)=v₀ from initial position s(0)=s₀.

a. For the following values of v₀ and s₀, find the position and velocity functions for all times at which the object is above the ground (s = 0).

v₀ = 49 m/s, s₀ = 60 m

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교과서 질문

{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm. 

a. Find the solution of the initial value problem.

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29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.


a. Find the approximations to y(0.2) and y(0.4) using Euler’s method with time steps of Δt = 0.2, 0.1, 0.05, and 0.025.


y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²

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{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation

dP/dt=kP(1−P/A),P0=P_0,

where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.

a. Find the solution of the initial value problem, for t≥0, in terms of K, A, and P0.

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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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