Domanda del libro di testo
33–42. Solving initial value problems Solve the following initial value problems.
y''(t) = teᵗ, y(0) = 0, y'(0) = 1
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 9, Problema 9.1.42
33–42. Solving initial value problems Solve the following initial value problems.
p'(x) = 2/(x² + x), p(1) = 0
Guida verificata passo dopo passo1
Identify the given differential equation and initial condition: \(p'(x) = \frac{2}{x^{2} + x}\) with \(p(1) = 0\).
Rewrite the derivative notation as \(\frac{dp}{dx} = \frac{2}{x^{2} + x}\) to recognize it as a separable differential equation.
Simplify the denominator by factoring: \(x^{2} + x = x(x + 1)\), so the equation becomes \(\frac{dp}{dx} = \frac{2}{x(x + 1)}\).
Integrate both sides with respect to \(x\): \(p(x) = \int \frac{2}{x(x + 1)} \, dx + C\), where \(C\) is the constant of integration.
Use partial fraction decomposition to express \(\frac{2}{x(x + 1)}\) as \(\frac{A}{x} + \frac{B}{x + 1}\), find \(A\) and \(B\), then integrate each term separately. Finally, apply the initial condition \(p(1) = 0\) to solve for \(C\).
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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 technique simplifies solving by integrating each side independently.
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06:06
Solving Separable Differential Equations
Integration of Rational Functions
Integrating rational functions often involves techniques like partial fraction decomposition to rewrite the integrand into simpler fractions. This method is essential for integrating expressions like 2/(x² + x) effectively.
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6:04Intro to Rational Functions
Initial Value Problems (IVP)
An initial value problem specifies a differential equation along with a condition at a particular point, such as p(1) = 0. Solving an IVP involves finding the general solution and then using the initial condition to determine the specific constant.
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Percorso guidato
05:03Initial Value Problems
Pratica correlata
Domanda del libro di testo
45–46. Harvesting problems Consider the harvesting problem in Example 6.
If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?
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Domanda del libro di testo
11–16. Initial value problems Solve the following initial value problems.
y'(t) − 3y = 12, y(1) = 4
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Domanda del libro di testo
17–18. {Use of Tech} Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function.
The population increases from 50 to 60 in the first month and eventually levels off at 150.
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Domanda del libro di testo
Explain how to solve a separable differential equation of the form
g(t)y'(y) = h(t)
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Domanda del libro di testo
27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.
An iron rod is removed from a blacksmith’s forge at a temperature of 900°C . Assume k=0.02 and the rod cools in a room with a temperature of 30°C When does the temperature of the rod reach 100°C?
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