Textbook Question
In Exercises 1–22, solve the differential equation.
e⁻ˣ dy + (e⁻ˣ y - 4x) dx = 0
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.20
In Exercises 1–22, solve the differential equation.
y' + 3x²y = 7x²
Verified step by step guidance1
Identify the type of differential equation. The given equation is a first-order linear differential equation of the form \(y' + P(x)y = Q(x)\), where \(P(x) = 3x^{2}\) and \(Q(x) = 7x^{2}\).
Find the integrating factor (IF), which is given by \(\mu(x) = e^{\int P(x) \, dx}\). Calculate \(\int 3x^{2} \, dx\) to determine the integrating factor.
Multiply both sides of the differential equation by the integrating factor \(\mu(x)\) to make the left side an exact derivative of the product \(\mu(x)y\).
Rewrite the left side as \(\frac{d}{dx}[\mu(x)y]\) and set it equal to the right side multiplied by \(\mu(x)\). This gives \(\frac{d}{dx}[\mu(x)y] = \mu(x) Q(x)\).
Integrate both sides with respect to \(x\) to find \(\mu(x)y = \int \mu(x) Q(x) \, dx + C\). Finally, solve for \(y\) by dividing both sides by \(\mu(x)\).
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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). It can be solved using an integrating factor, which simplifies the equation into an exact derivative, allowing integration to find the solution.
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Integrating Factor Method
The integrating factor is typically e^(∫P(x)dx). Multiplying the entire differential equation by this factor transforms the left side into the derivative of (integrating factor × y), enabling straightforward integration to solve for y.
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Integration of Functions Involving Polynomials
Solving the equation requires integrating expressions involving polynomials like x². Understanding basic integration rules for powers of x is essential to compute the integrating factor and the integral of the right-hand side.
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Related Practice
Textbook Question
Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.
(x²+y²)dx + xy dy = 0
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Textbook Question
In Exercises 1–22, solve the differential equation.
xy' + 2y = 1 - x⁻¹
Textbook Question
In Exercises 1–22, solve the differential equation.
y'/2 + y = e⁻ˣ sin x
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.
x dy - (x⁴ - y) dx = 0