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Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 9 - First-Order Differential EquationsProblem 9.1.17
Chapter 9, Problem 9.1.17

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' = 2xy + 2y, y(0) = 3, dx = 0.2

Verified step by step guidance
1
Identify the differential equation and initial condition: \(y' = 2xy + 2y\) with \(y(0) = 3\). Here, \(y'\) represents the derivative of \(y\) with respect to \(x\).
Set the step size \(\Delta x = 0.2\) and initialize the first point as \((x_0, y_0) = (0, 3)\).
Use Euler's method formula to find the next value: \(y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x\), where \(f(x, y) = 2xy + 2y\) is the right-hand side of the differential equation.
Calculate the first three approximations by iterating the formula: - For \(n=0\), compute \(y_1\) using \(x_0\) and \(y_0\). - For \(n=1\), compute \(y_2\) using \(x_1 = x_0 + \Delta x\) and \(y_1\). - For \(n=2\), compute \(y_3\) using \(x_2 = x_1 + \Delta x\) and \(y_2\).
Find the exact solution by solving the differential equation analytically (e.g., using an integrating factor or separation of variables), then compare the exact values at \(x = 0.2, 0.4, 0.6\) with your Euler approximations to assess 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 over steps.
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Euler's Method

Initial Value Problems (IVP)

An initial value problem specifies a differential equation along with a starting point (initial condition) for the solution. The goal is to find a function that satisfies both the differential equation and the initial condition, ensuring a unique solution that can be approximated or solved exactly.
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Initial Value Problems

Exact Solution and Error Analysis

The exact solution is the precise function satisfying the differential equation and initial condition, often found analytically. Comparing Euler's approximations to the exact solution helps assess accuracy, revealing how step size and method limitations affect error magnitude.
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Determining Error and Relative Error
Related Practice
Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


(1+x)y' + y = √x

Textbook Question

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.


y = 1 + ∫₀ ͯ y(t) dt

Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


(t-1)³ ds/dt + 4(t-1)²s = t+1, t >1

Textbook Question

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.


(x+1) dy/dx - 2 (x² + x)y = exp(x²) / (x+1), x > -1, y(0) = 5

Textbook Question

Solve the Bernoulli equations in Exercises 29–32.


x²y' + 2xy = y³

Textbook Question

Show that the solution of the initial value problem

y' = x + y, y(x₀) = y₀

is

y = -1 -x + (1 + x₀ + y₀) exp(x-x₀).