Textbook Question
In Exercises 1–22, solve the differential equation.
xy' - y = 2x ln x
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 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
Chapter 9, Problem 9.PE.10
In Exercises 1–22, solve the differential equation.
y'/2 + y = e⁻ˣ sin x
Verified step by step guidance1
Rewrite the given differential equation to isolate \( y' \). Starting with \( \frac{y'}{2} + y = e^{-x} \sin x \), multiply both sides by 2 to get \( y' + 2y = 2 e^{-x} \sin x \).
Recognize that this is a first-order linear differential equation of the form \( y' + P(x) y = Q(x) \), where \( P(x) = 2 \) and \( Q(x) = 2 e^{-x} \sin x \).
Find the integrating factor \( \mu(x) \) using the formula \( \mu(x) = e^{\int P(x) \, dx} = e^{\int 2 \, dx} = e^{2x} \).
Multiply the entire differential equation by the integrating factor \( e^{2x} \) to obtain \( e^{2x} y' + 2 e^{2x} y = 2 e^{2x} e^{-x} \sin x \), which simplifies to \( \frac{d}{dx} \left( e^{2x} y \right) = 2 e^{x} \sin x \).
Integrate both sides with respect to \( x \): \( \int \frac{d}{dx} \left( e^{2x} y \right) dx = \int 2 e^{x} \sin x \, dx \). This gives \( e^{2x} y = \int 2 e^{x} \sin x \, dx + C \), where \( C \) is the constant of integration.
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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
A first-order linear differential equation has the form y' + P(x)y = Q(x). Solving it involves finding an integrating factor to simplify the equation into an exact derivative, allowing integration to find the general solution.
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Integrating Factor Method
The integrating factor is a function, usually e^(∫P(x)dx), used to multiply both sides of a linear differential equation. This transforms the left side into the derivative of a product, making the equation easier to integrate and solve.
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Solving Nonhomogeneous Equations with Exponential and Trigonometric Terms
When the nonhomogeneous term involves expressions like e^(-x)sin(x), solutions often require using the method of undetermined coefficients or variation of parameters. Recognizing the form helps in guessing a particular solution that fits the right-hand side.
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Related Practice
Textbook Question
In Exercises 1–22, solve the differential equation.
e⁻ˣ dy + (e⁻ˣ y - 4x) dx = 0
Textbook Question
In Exercises 1–22, solve the differential equation.
xy' + 2y = 1 - x⁻¹
Textbook Question
In Exercises 43 and 44, let S represent the pounds of salt in a tank at time t minutes. Set up a differential equation representing the given information and the rate at which S changes. Then solve for S and answer the particular questions.
Pure water flows into a tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 5 gal/min. The tank initially holds 200 gal of solution containing 50 pounds of salt.
a. How many gallons of solution are in the tank after 1 minute? after 10 minutes? after 200 minutes?
Textbook Question
In Exercises 1–22, solve the differential equation.
y' + 3x²y = 7x²
Textbook Question
In Exercises 1–22, solve the differential equation.
x dy + (3y - x⁻² cos x) dx = 0, x > 0