Textbook Question
9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.
Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 9, Problem 9.1.42
33–42. Solving initial value problems Solve the following initial value problems.
p'(x) = 2/(x² + x), p(1) = 0
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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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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Related Practice
Textbook Question
33–42. Solving initial value problems Solve the following initial value problems.
y''(t) = teᵗ, y(0) = 0, y'(0) = 1
Textbook Question
21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
y'(t) = 3 + e⁻²ᵗ
Textbook Question
17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = eᵗʸ, y(0) = 1
Textbook Question
Explain why the graph of the solution to the initial value problem y'(t) = t²/(1 - t), y(-1) = ln 2 cannot cross the line t = 1.
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Textbook Question
17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = yeᵗ, y(0) = −1
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