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.R.14
Capitolo 9, Problema 9.R.14

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.
y′(x) = x/y, y(2) = 4

Guida verificata passo dopo passo
1
Identify the type of differential equation given: \( y'(x) = \frac{x}{y} \). This is a separable differential equation because the right side can be expressed as a function of \( x \) divided by a function of \( y \).
Rewrite the differential equation by separating variables: multiply both sides by \( y \) and multiply both sides by \( dx \) to get \( y \, dy = x \, dx \).
Integrate both sides: integrate \( y \, dy \) with respect to \( y \) and \( x \, dx \) with respect to \( x \). This gives \( \int y \, dy = \int x \, dx \).
After integrating, you will have an implicit solution involving \( y \) and \( x \) plus a constant of integration \( C \). Use the initial condition \( y(2) = 4 \) to solve for \( C \).
Finally, write the explicit solution for \( y \) in terms of \( x \) by solving the implicit equation for \( y \), if possible.

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:
3m

Concetti chiave

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

Separable Differential Equations

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

Initial Value Problems (IVP)

An initial value problem specifies a differential equation along with a condition that the solution must satisfy at a particular point. This condition helps determine the unique solution by solving for the constant of integration after finding the general solution.
Video consigliato:
Percorso guidato
05:03
Initial Value Problems

Integration Techniques

Solving separable equations requires integrating both sides after separating variables. Familiarity with basic integration rules and techniques, such as power rule and substitution, is essential to find the explicit form of the solution.
Video consigliato:
Percorso guidato
06:18
Integration by Parts for Definite Integrals
Pratica correlata
Domanda del libro di testo

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.

c. When does the temperature of the coffee reach 50°C?

81
views
Domanda del libro di testo

Direction fields Consider the direction field for the equation y′=y(2−y) shown in the figure and initial conditions of the form y(0)=A.

a. Sketch a solution on the direction field with the initial condition y(0)=1.

60
views
Domanda del libro di testo

Logistic growth The population of a rabbit community is governed by the initial value problem

P′(t) = 0.2 P (1 − P/1200), P(0) = 50

a. Find the equilibrium solutions.

47
views
Domanda del libro di testo

2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.

y′(t) = (2t+1)(y²+1)

44
views
Domanda del libro di testo

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.

y′(x) = 4x csc y, y(0) = π/2

78
views
Domanda del libro di testo

Euler’s method Consider the initial value problem y′(t)=1/2y,y(0)=1. 

b. Use Euler’s method with Δt=0.05 to compute approximations to y(0.1) and y(0.2). 

51
views