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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.12
Capitolo 9, Problema 9.R.12

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.
y′(t) = -3y + 9, y(0) = 4

Guida verificata passo dopo passo
1
Identify the type of differential equation given: \( y'(t) = -3y + 9 \) is a first-order linear ordinary differential equation.
Rewrite the equation in standard linear form: \( y' + 3y = 9 \). This helps us recognize the integrating factor method.
Calculate the integrating factor \( \mu(t) \) using the formula \( \mu(t) = e^{\int P(t) dt} \), where \( P(t) = 3 \). So, \( \mu(t) = e^{3t} \).
Multiply both sides of the differential equation by the integrating factor \( e^{3t} \) to get \( e^{3t} y' + 3 e^{3t} y = 9 e^{3t} \). Notice the left side is the derivative of \( e^{3t} y \).
Integrate both sides with respect to \( t \): \( \int \frac{d}{dt} (e^{3t} y) dt = \int 9 e^{3t} dt \). Then solve for \( y(t) \) and apply the initial condition \( y(0) = 4 \) to find the constant of integration.

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Concetti chiave

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First-Order Linear Differential Equations

These are differential equations involving the first derivative of a function and the function itself, typically in the form y' + p(t)y = q(t). Solving them often involves finding an integrating factor or using other methods to obtain the general solution.
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07:39
Classifying Differential Equations

Initial Value Problems (IVPs)

An IVP specifies the value of the unknown function at a particular point, such as y(0) = 4. This condition allows us to find a unique solution to the differential equation by determining the constant of integration.
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Percorso guidato
05:03
Initial Value Problems

Method of Integrating Factor

This technique solves linear differential equations by multiplying both sides by an integrating factor, usually e^(∫p(t)dt), which simplifies the equation into an exact derivative. Integrating then yields the solution.
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07:33
Euler's Method
Pratica correlata
Domanda del libro di testo

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

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

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Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.

e. Discuss some possible shortcomings of this model. Why might the carrying capacity be either greater than or less than the value predicted by the model?

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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.

d. For what values of A are the corresponding solutions decreasing, for t≥0

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Domanda del libro di testo

A predator-prey model Consider the predator-prey model

x′(t) = −4x + 2xy, y′(t) = 5y − xy

c. Find the equilibrium points for the system.

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Domanda del libro di testo

Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.

b. Use the direction field to sketch the solution curve that passes through the point (0,−1/2).

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7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁t⁵ + C₂t⁻⁴ - t³; t²u''(t) - 20u(t) = 14t³

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