Textbook Question
First-Order Linear Equations
Solve the differential equations in Exercises 1–14.
tan θ dr/dθ + r = sin²θ, 0 < θ < π/2
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 9.1.39
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
Verified step by step guidance1
Identify the differential equation and initial condition: \(y' = 2x \exp(x^2)\) with \(y(0) = 2\).
Set up Euler's method formula: \(y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x\), where \(f(x, y) = 2x \exp(x^2)\) and \(\Delta x = 0.1\).
Calculate the number of steps needed to reach \(x^* = 1\) starting from \(x_0 = 0\) using \(\text{steps} = \frac{x^* - x_0}{\Delta x} = \frac{1 - 0}{0.1} = 10\) steps.
Iteratively apply Euler's method: for each step \(n\) from 0 to 9, compute \(y_{n+1} = y_n + 2x_n \exp(x_n^2) \cdot 0.1\) and update \(x_{n+1} = x_n + 0.1\).
To find the exact solution at \(x = 1\), integrate the differential equation: \(y = \int 2x \exp(x^2) \, dx + C\). Use substitution \(u = x^2\) to solve the integral, then apply the initial condition \(y(0) = 2\) to find \(C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Euler's Method
Euler's method is a numerical technique to approximate solutions of first-order differential equations. Starting from an initial condition, it uses the slope given by the differential equation to estimate the next value by moving in small steps (dx). This iterative process continues until reaching the desired point.
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Euler's Method
Initial Value Problem (IVP)
An initial value problem specifies a differential equation along with a starting point (initial condition) for the solution. The solution curve is uniquely determined by this initial value, allowing numerical methods like Euler's to approximate the solution over an interval.
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05:03Initial Value Problems
Exact Solution of Differential Equations
The exact solution is an explicit formula that satisfies the differential equation and initial condition. Finding it often involves integration or known solution techniques. Comparing the exact solution to numerical approximations helps assess the accuracy of methods like Euler's.
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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
Carbon monoxide pollution An executive conference room of a corporation contains 4500 ft³ of air initially free of carbon monoxide. Starting at time t = 0, cigarette smoke containing 4% carbon monoxide is blown into the room at the rate of 0.3 ft³/min. A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0.3 ft³/min. Find the time when the concentration of carbon monoxide in the room reaches 0.01%.
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Textbook Question
Using Euler’s Method
In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.
y' = x(1-y), y(1) = 0, dx = 0.2
Textbook Question
Use Euler’s method with dx = 1/3 to estimate y(2) if y′ = x sin y and y(0) = 1. What is the exact value of y(2)?
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' = 1 + y², y(0) = 0, dx = 0.1, x* = 1
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