Skip to main content
Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 9 - Differential EquationsProblema 9.4.6
Capitolo 9, Problema 9.4.6

5–10. First-order linear equations Find the general solution of the following equations.


y'(x) = −y + 2

Guida verificata passo dopo passo
1
Rewrite the differential equation in the standard linear form: \(y'(x) + y = 2\).
Identify the integrating factor \(\mu(x)\), which is given by \(\mu(x) = e^{\int 1 \, dx} = e^{x}\).
Multiply both sides of the equation by the integrating factor to get: \(e^{x} y' + e^{x} y = 2 e^{x}\).
Recognize that the left side is the derivative of the product \(e^{x} y\), so write it as \(\frac{d}{dx} (e^{x} y) = 2 e^{x}\).
Integrate both sides with respect to \(x\): \(\int \frac{d}{dx} (e^{x} y) \, dx = \int 2 e^{x} \, dx\), then solve for \(y\) by dividing by \(e^{x}\) and adding the constant of integration.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
1m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

First-Order Linear Differential Equations

A first-order linear differential equation has the form y' + p(x)y = q(x). It involves the first derivative of the unknown function and can be solved using integrating factors or other standard methods. Recognizing this form is essential to apply the correct solution technique.
Video consigliato:
07:39
Classifying Differential Equations

Integrating Factor Method

The integrating factor method involves multiplying the entire differential equation by a specially chosen function, usually e^(∫p(x)dx), to rewrite the left side as a derivative of a product. This simplifies solving the equation by allowing direct integration.
Video consigliato:
07:33
Euler's Method

General Solution of Differential Equations

The general solution includes all possible solutions of a differential equation and typically contains an arbitrary constant. It combines the homogeneous solution (solving y' + p(x)y = 0) and a particular solution to the nonhomogeneous equation.
Video consigliato:
04:00
Solutions to Basic Differential Equations
Pratica correlata
Domanda del libro di testo

33–38. {Use of Tech} Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem.

y'(t) = 2t²/(y² − 1), y(0) = 0

41
views
Domanda del libro di testo

21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.

P′(t) = 0.05P − 0.001P²; P(0) = 10, P(0) = 40, P(0) = 80

44
views
Domanda del libro di testo

21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.

P′(t) = 0.05P(1−P/800); P(0) = 100, P(0) = 400, P(0) = 700

57
views
Domanda del libro di testo

Consider the differential equation y'(t) = t² - 3y² and the solution curve that passes through the point (3, 1). What is the slope of the curve at (3, 1)?

53
views
Domanda del libro di testo

What is the equilibrium solution of the equation y'(t) = 3y − 9? Is it stable or unstable?

64
views
Domanda del libro di testo

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

e⁴ᵗy'(t) = 5

65
views