Textbook Question
Use Euler’s method with dx = 0.2 to estimate y(2) if y′ = y/x and y(1) = 2. What is the exact value of y(2)?
Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 9, Problem 9.2.15
Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.
dy/dt + 2y = 3, y(0) = 1
Verified step by step guidance1
Identify the type of differential equation. The given equation is a first-order linear ordinary differential equation of the form \(\frac{dy}{dt} + P(t)y = Q(t)\), where \(P(t) = 2\) and \(Q(t) = 3\).
Find the integrating factor \(\mu(t)\) using the formula \(\mu(t) = e^{\int P(t)\,dt}\). In this case, calculate \(\mu(t) = e^{\int 2\,dt}\).
Multiply both sides of the differential equation by the integrating factor \(\mu(t)\) to rewrite the left side as the derivative of a product: \(\frac{d}{dt}[\mu(t) y] = \mu(t) Q(t)\).
Integrate both sides with respect to \(t\) to find \(\mu(t) y = \int \mu(t) Q(t)\, dt + C\), where \(C\) is the constant of integration.
Use the initial condition \(y(0) = 1\) to solve for the constant \(C\), then solve for \(y(t)\) by dividing both sides by \(\mu(t)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
First-Order Linear Differential Equations
These are differential equations of the form dy/dt + P(t)y = Q(t), where the solution involves finding an integrating factor to simplify the equation. Recognizing this form allows the use of systematic methods to solve for y(t).
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Classifying Differential Equations
Integrating Factor Method
This technique involves multiplying the entire differential equation by an integrating factor, usually e^(∫P(t)dt), to rewrite the left side as a derivative of a product. This simplifies solving the equation by enabling direct integration.
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07:33Euler's Method
Initial Value Problems (IVP)
An IVP specifies the value of the unknown function at a particular point, such as y(0) = 1. This condition is used to find the unique solution to the differential equation that satisfies the initial condition.
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05:03Initial Value Problems
Related Practice
Textbook Question
Use Euler’s method with dx = 0.5 to estimate y(5) if y′ = y²/√x and y(1) = −1. What is the exact value of y(5)?
Textbook Question
First-Order Linear Equations
Solve the differential equations in Exercises 1–14.
xdy/dx + y = e ͯ, x > 0
Textbook Question
Solve the Bernoulli equations in Exercises 29–32.
y' - y = xy²
Textbook Question
What integral equation is equivalent to the initial value problem y' = f(x), y(x₀) = y₀?
Textbook Question
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' = 2y/x, y(1) = -1, dx = 0.5
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