Textbook Question
Integral Equations
In Exercises 7–12, write an equivalent first-order differential equation
and initial condition for y.
y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 9, Problem 9.2.14
First-Order Linear Equations
Solve the differential equations in Exercises 1–14.
tan θ dr/dθ + r = sin²θ, 0 < θ < π/2
Verified step by step guidance1
Rewrite the given differential equation in the standard linear form. The original equation is \(\tan \theta \frac{dr}{d\theta} + r = \sin^{2} \theta\). Divide both sides by \(\tan \theta\) to isolate \(\frac{dr}{d\theta}\), resulting in \(\frac{dr}{d\theta} + \frac{r}{\tan \theta} = \frac{\sin^{2} \theta}{\tan \theta}\).
Identify the integrating factor (IF) for the linear differential equation \(\frac{dr}{d\theta} + P(\theta) r = Q(\theta)\), where \(P(\theta) = \frac{1}{\tan \theta}\). The integrating factor is given by \(\mu(\theta) = e^{\int P(\theta) d\theta}\).
Calculate the integral \(\int \frac{1}{\tan \theta} d\theta\). Recall that \(\frac{1}{\tan \theta} = \cot \theta\), so the integral becomes \(\int \cot \theta d\theta\). Use the known integral formula for \(\cot \theta\).
Multiply the entire differential equation by the integrating factor \(\mu(\theta)\) to write the left side as the derivative of the product \(\mu(\theta) r\). This transforms the equation into \(\frac{d}{d\theta} [\mu(\theta) r] = \mu(\theta) Q(\theta)\).
Integrate both sides with respect to \(\theta\) to find \(\mu(\theta) r = \int \mu(\theta) Q(\theta) d\theta + C\). Finally, solve for \(r(\theta)\) by dividing both sides by \(\mu(\theta)\).
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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
These are differential equations of the form dy/dx + P(x)y = Q(x), where the highest derivative is first order and the equation is linear in the unknown function and its derivative. Solving them typically involves finding an integrating factor to simplify the equation into an exact derivative.
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Classifying Differential Equations
Integrating Factor Method
An integrating factor is a function, usually denoted μ(x), used to multiply a linear differential equation to make the left side an exact derivative. It is found by μ(x) = e^(∫P(x)dx), enabling straightforward integration and solution of the equation.
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Trigonometric Functions and Their Properties
Understanding the behavior and identities of trigonometric functions like tan(θ) and sin²(θ) is essential. These functions often appear in differential equations involving angular variables, and their properties help simplify and integrate terms effectively.
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Related Practice
Textbook Question
In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.
y′ = √x/y, y > 0, y(0) = 1, dx = 0.1, x* = 1
Textbook Question
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Textbook Question
In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.
y' = 2xexp(x²) , y(0) = 2, dx = 0.1, x* = 1
Textbook Question
Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.
Textbook Question
Use Euler’s method with dx = 0.2 to estimate y(1) if y′ = y and y(0) = 1. What is the exact value of y(1)?