Ch. 9 - First-Order Differential Equations

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 9 - First-Order Differential Equations

Problem 9.1.24

Chapter 9, Problem 9.1.24

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)?

Verified step by step guidance

Identify the differential equation and initial condition: \(\frac{dy}{dx} = x \sin y\) with \(y(0) = 1\).

Set the step size \(\Delta x = \frac{1}{3}\) and determine the number of steps needed to reach \(x = 2\). Since \(2 \div \frac{1}{3} = 6\), you will perform 6 steps.

Apply Euler's method iteratively using the formula: \(y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n)\), where \(f(x, y) = x \sin y\). Start with \(x_0 = 0\) and \(y_0 = 1\).

For each step, calculate \(y_{n+1}\) by plugging in the current values of \(x_n\) and \(y_n\) into the derivative function, then update \(x_{n+1} = x_n + \Delta x\).

To find the exact value of \(y(2)\), solve the differential equation analytically by separating variables or using an appropriate method, then apply the initial condition to find the constant of integration.

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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. It uses a step size (dx) to incrementally estimate the value of the function by moving along the slope given by the derivative. This method is especially useful when an exact solution is difficult to find.

Solving Initial Value Problems

An initial value problem specifies the value of the unknown function at a starting point, allowing the differential equation to be solved uniquely. Here, y(0) = 1 provides the initial condition needed to apply Euler's method and to find the exact solution.

Exact Solution of Differential Equations

The exact solution involves finding an explicit formula for y in terms of x that satisfies the differential equation and initial condition. This often requires techniques like separation of variables or integrating factors, enabling comparison with numerical approximations.

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Related Practice

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

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

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

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

e²ˣy' + 2e²ˣ y = 2x

Textbook Question

Is either of the following equations correct? Give reasons for your answers.

a. (1/cosx) ∫ cos x dx = tan x + C

b. (1/cosx) ∫ cos x dx = tan x + C / cos x

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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' = 1 + y², y(0) = 0, dx = 0.1, x* = 1

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