Domanda del libro di testo
In Exercises 1–22, solve the differential equation.
dy + x(2y - e^(x-x²))dx = 0
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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.PE.21
In Exercises 1–22, solve the differential equation.
y' = xy ln x ln y
Guida verificata passo dopo passo1
Rewrite the given differential equation \(y' = xy \ln x \ln y\) in Leibniz notation as \(\frac{dy}{dx} = xy \ln x \ln y\).
Separate the variables by dividing both sides to isolate \(y\) terms on one side and \(x\) terms on the other: write it as \(\frac{1}{y \ln y} dy = x \ln x \, dx\).
Integrate both sides separately: compute \(\int \frac{1}{y \ln y} dy\) on the left and \(\int x \ln x \, dx\) on the right.
For the left integral, use substitution \(u = \ln y\) which implies \(du = \frac{1}{y} dy\), transforming the integral into \(\int \frac{1}{u} du\).
For the right integral, use integration by parts where you let \(u = \ln x\) and \(dv = x \, dx\), then apply the formula \(\int u \, dv = uv - \int v \, du\).
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Separable Differential Equations
A separable differential equation can be written as a product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the solution.
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Solving Separable Differential Equations
Properties of Logarithmic Functions
Understanding the natural logarithm function, ln(x), is essential, including its domain (x > 0) and its behavior. Logarithmic properties help simplify expressions and manipulate terms involving ln(x) and ln(y) during the solution process.
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Percorso guidato
06:21Properties of Functions
Integration Techniques
Solving the separated equation requires integrating functions involving logarithms and polynomials. Familiarity with integration methods, such as substitution and integration by parts, is important to evaluate the integrals correctly.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
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Domanda del libro di testo
In Exercises 1–22, solve the differential equation.
y' = xeˣ⁻ʸ csc y
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Domanda del libro di testo
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.
c. When will the tank have exactly 5 pounds of salt and how many gallons of solution will be in the tank?
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Domanda del libro di testo
In Exercises 23–28, solve the initial value problem.
x dy/dx + 2y = x² + 1, x > 0, y(1) = 1
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Domanda del libro di testo
In Exercises 1–22, solve the differential equation.
(1+eˣ) dy + (yeˣ + e⁻ˣ) dx = 0
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Domanda del libro di testo
In Exercises 1–22, solve the differential equation.
y' = eʸ/xy
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