Domanda del libro di testo
Integral Equations
In Exercises 7–12, write an equivalent first-order differential equation
and initial condition for y.
y = ∫₁ ͯ 1/t dt
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Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 9, Problema 9.2.4
First-Order Linear Equations
Solve the differential equations in Exercises 1–14.
y' + (tanx)y = cos²x, -π/2 < x < π/2
Guida verificata passo dopo passo1
Identify the given differential equation as a first-order linear differential equation of the form \(y' + P(x)y = Q(x)\), where \(P(x) = \tan x\) and \(Q(x) = \cos^{2} x\).
Find the integrating factor \(\mu(x)\) using the formula \(\mu(x) = e^{\int P(x) \, dx} = e^{\int \tan x \, dx}\).
Compute the integral \(\int \tan x \, dx\) to determine the explicit form of the integrating factor \(\mu(x)\).
Multiply both sides of the original differential equation by the integrating factor \(\mu(x)\) to rewrite the left side as the derivative of the product \(\mu(x) y\).
Integrate both sides with respect to \(x\) to find \(\mu(x) y = \int \mu(x) Q(x) \, dx + C\), then solve for \(y\) by dividing both sides by \(\mu(x)\).
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First-Order Linear Differential Equations
These are differential equations of the form y' + P(x)y = Q(x), where y' is the derivative of y with respect to x. They can be solved using an integrating factor, which simplifies the equation into an exact derivative, allowing integration to find the solution.
Video consigliato:
07:39
Classifying Differential Equations
Integrating Factor Method
The integrating factor is a function, usually denoted μ(x) = e^(∫P(x)dx), used to multiply both sides of a linear differential equation. This transforms the left side into the derivative of (μ(x)y), making it easier to integrate and solve for y.
Video consigliato:
07:33Euler's Method
Trigonometric Functions in Differential Equations
Understanding the properties and integrals of trigonometric functions like tan(x) and cos²(x) is essential. These functions appear in P(x) and Q(x), and their integrals determine the integrating factor and the particular solution of the differential equation.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo
First-Order Linear Equations
Solve the differential equations in Exercises 1–14.
(1+x)y' + y = √x
49
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Domanda del libro di testo
Integral Equations
In Exercises 7–12, write an equivalent first-order differential equation
and initial condition for y.
y = 1 + ∫₀ ͯ y(t) dt
23
views
Domanda del libro di testo
Using Euler’s Method
In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.
y' = 2xy + 2y, y(0) = 3, dx = 0.2
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Domanda del libro di testo
First-Order Linear Equations
Solve the differential equations in Exercises 1–14.
xdy/dx + y = e ͯ, x > 0
16
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