Domanda del libro di testo
In Exercises 23–28, solve the initial value problem.
y dx + (3x - xy + 2)dy = 0, y(2) = -1, y < 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.17
In Exercises 1–22, solve the differential equation.
y' = sin³ x cos² y
Guida verificata passo dopo passo1
Recognize that the given differential equation is separable since it can be written as a product of a function of x and a function of y: \(y' = \sin^{3} x \cos^{2} y\).
Rewrite the differential equation in differential form: \(\frac{dy}{dx} = \sin^{3} x \cos^{2} y\).
Separate the variables by dividing both sides by \(\cos^{2} y\) and multiplying both sides by \(dx\): \(\frac{dy}{\cos^{2} y} = \sin^{3} x \, dx\).
Express the left side in terms of \(\sec^{2} y\) to facilitate integration: \(\sec^{2} y \, dy = \sin^{3} x \, dx\).
Integrate both sides separately: \(\int \sec^{2} y \, dy = \int \sin^{3} x \, dx\), then solve each integral using appropriate techniques (such as substitution or trigonometric identities).
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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 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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06:06
Solving Separable Differential Equations
Integration of Trigonometric Functions
Solving the equation involves integrating powers of sine and cosine functions. Familiarity with trigonometric identities and integration techniques, such as substitution or power-reduction formulas, is essential to evaluate these integrals correctly.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Implicit Solutions and Initial Conditions
After integration, solutions may be implicit, involving both x and y in an equation. Understanding how to interpret implicit solutions and apply initial conditions, if given, helps in finding explicit solutions or particular solutions to the differential equation.
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Percorso guidato
05:03Initial Value Problems
Pratica correlata
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.
b. How many pounds of salt are in the tank after 1 minute? after 30 minutes?
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Domanda del libro di testo
In Exercises 1–22, solve the differential equation.
(x + 3y²) dy + y dx = 0 (Hint: d(xy) = y dx + x dy)
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Domanda del libro di testo
In Exercises 1–22, solve the differential equation.
2y' - y = xe^(x/2)
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Domanda del libro di testo
In Exercises 1–22, solve the differential equation.
y' = (y²-1)x⁻¹
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Domanda del libro di testo
In Exercises 1–22, solve the differential equation.
sec x dy + x cos² y dx = 0
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