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 9.1.42

Chapter 9, Problem 9.1.42

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

Verified step by step guidance

Identify the differential equation and initial condition: \(y' = 1 + y^2\) with \(y(0) = 0\).

Set the step size \(\Delta x = 0.1\) and the target point \(x^* = 1\). Determine the number of steps needed: \(n = \frac{x^* - 0}{\Delta x} = 10\) steps.

Apply Euler's method iteratively using the formula: \(y_{k+1} = y_k + \Delta x \cdot f(x_k, y_k)\), where \(f(x, y) = 1 + y^2\). Start with \(y_0 = 0\) at \(x_0 = 0\).

For each step \(k\) from 0 to 9, compute \(y_{k+1}\) using the previous value \(y_k\) and the slope \(f(x_k, y_k)\). Update \(x_{k+1} = x_k + \Delta x\) accordingly.

To find the exact solution at \(x^* = 1\), solve the initial value problem analytically. Recognize that \(y' = 1 + y^2\) is separable, so separate variables and integrate: \(\int \frac{dy}{1 + y^2} = \int dx\). Then apply the initial condition to find the constant of integration and evaluate \(y(1)\).

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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 to incrementally estimate the function's value by moving along the slope given by the differential equation. This method is especially useful when an exact solution is difficult to find.

Initial Value Problems (IVP)

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) = 0 sets the initial condition, which is essential for applying Euler's method and finding the particular solution.

Exact Solution of Differential Equations

The exact solution is an explicit formula that satisfies the differential equation and initial condition. For y' = 1 + y², the exact solution involves recognizing it as a separable equation, often leading to a trigonometric function like tangent, which can be evaluated at x* for comparison with the numerical estimate.

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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' = 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

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

Using Euler’s Method

y' = y²(1+2x), (y-1) = 1, dx = 0.5

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.