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 9.1.18

Chapter 9, Problem 9.1.18

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' = y²(1+2x), (y-1) = 1, dx = 0.5

Verified step by step guidance

Identify the initial value problem (IVP) and the given parameters: the differential equation is \(y' = y^{2}(1 + 2x)\), the initial condition is \(y(1) = 1\), and the step size is \(\Delta x = 0.5\).

Set up Euler's method formula: given a current point \((x_n, y_n)\), the next approximation is \(y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n)\), where \(f(x, y) = y^{2}(1 + 2x)\) is the derivative function.

Calculate the first approximation \(y_1\) at \(x_1 = x_0 + \Delta x = 1 + 0.5 = 1.5\) using the initial values: \(y_1 = y_0 + 0.5 \times y_0^{2}(1 + 2x_0)\).

Calculate the second approximation \(y_2\) at \(x_2 = 2.0\) using the previous approximation: \(y_2 = y_1 + 0.5 \times y_1^{2}(1 + 2x_1)\).

Calculate the third approximation \(y_3\) at \(x_3 = 2.5\) similarly: \(y_3 = y_2 + 0.5 \times y_2^{2}(1 + 2x_2)\). After these, find the exact solution of the differential equation with the initial condition and compare the values to analyze the accuracy.

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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 value, it uses the slope given by the differential equation to estimate the function's value at successive points, incrementing by a fixed step size. This method is simple but can accumulate errors depending on the step size.

Initial Value Problem (IVP)

An Initial Value Problem specifies a differential equation along with a starting point (initial condition) for the solution. The initial condition, such as y(1) = 1, provides a specific value that the solution must satisfy, allowing for a unique solution curve to be approximated or found exactly.

Exact Solution and Error Analysis

The exact solution is the precise function satisfying the differential equation and initial condition. Comparing Euler's approximations to the exact solution helps assess the accuracy of the numerical method. Error analysis involves calculating the difference between approximate and exact values, often influenced by step size and method order.

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Related Practice

Textbook Question

What integral equation is equivalent to the initial value problem y' = f(x), y(x₀) = y₀?

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' = 2y/x, y(1) = -1, dx = 0.5

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Textbook Question

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

dy/dx + xy = x, y(0) = -6

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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