Textbook Question
In Exercises 1–22, solve the differential equation.
y' = xeʸ√(x-2)
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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.12
In Exercises 1–22, solve the differential equation.
xy' - y = 2x ln x
Verified step by step guidance1
Rewrite the given differential equation in the standard linear form. Starting with \(xy' - y = 2x \ln x\), divide both sides by \(x\) (assuming \(x \neq 0\)) to get \(y' - \frac{1}{x} y = 2 \ln x\).
Identify the integrating factor \(\mu(x)\), which is given by \(\mu(x) = e^{\int P(x) \, dx}\) where \(P(x)\) is the coefficient of \(y\) in the standard form. Here, \(P(x) = -\frac{1}{x}\), so calculate \(\mu(x) = e^{\int -\frac{1}{x} \, dx}\).
Simplify the integrating factor \(\mu(x)\) by evaluating the integral \(\int -\frac{1}{x} \, dx\) and then exponentiating the result.
Multiply the entire differential equation by the integrating factor \(\mu(x)\) to transform the left side into the derivative of the product \(\mu(x) y\). This gives \(\frac{d}{dx} [\mu(x) y] = \mu(x) \cdot 2 \ln x\).
Integrate both sides with respect to \(x\) to find \(\mu(x) y = \int \mu(x) \cdot 2 \ln x \, dx + C\), where \(C\) is the constant of integration. 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). 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 denoted μ(x), used to multiply both sides of a linear differential equation to make the left side an exact derivative. It is typically μ(x) = e^(∫P(x) dx), facilitating straightforward integration.
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Properties of Logarithmic Functions
Understanding the natural logarithm function ln(x) is essential, especially its domain (x > 0) and differentiation properties. This knowledge helps in manipulating terms like 2x ln x and ensuring the solution respects the function's domain.
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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.
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.
sec x dy + x cos² y dx = 0
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Textbook Question
In Exercises 1–22, solve the differential equation.
x dy + (3y - x⁻² cos x) dx = 0, x > 0