### Using Euler's Method to Approximate Solutions to Differential EquationsEuler's method is a simple numerical approach to approximating solutions to ordinary differential equations (ODEs). Given the step size $ h $ and an initial value problem, the method estimates the values of the unknown function at discrete points.#### Problem StatementUse Euler's method with step size $ 0.5 $ to compute the approximate y-values $ y_1 $, $ y_2 $, $ y_3 $, and $ y_4 $ for the solution of the given initial-value problem:\[ y' = -2 + 2x + 4y, \quad y(1) = 2. \]#### Procedure1. **Initial Value**: - $ x_0 = 1 $ - $ y_0 = 2 $2. **Step Size**: - $ h = 0.5 $3. **Equations**: $ y' = f(x, y) = -2 + 2x + 4y $4. **Iteration Formula**: \[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \]Use the iteration formula to compute $ y_1, y_2, y_3, $ and $ y_4 $.#### Solution1. **First Iteration**: - $ x_0 = 1 $, $ y_0 = 2 $ - Compute $ f(x_0, y_0) $: \[ f(1, 2) = -2 + 2(1) + 4(2) = -2 + 2 + 8 = 8 \] - Calculate $ y_1 $: \[ y_1 = y_0 + h \cdot f(x_0, y_0) = 2 + 0.5 \cdot 8 = 2 + 4 = 6 \]2. **Second Iteration**: - $ x_1 = 1.5 $, $ y_1 = 6 $ - Compute $ f(x_1, y_1) $: \[ f(1.5, 6) = -2 + 2(1.5) +

Answered: Use Euler's method with step size 0.5…

Use Euler's method with step size 0.5 to compute the approximate y-values y1, Y2, Y3, and y4 of the solution of the initial-value problem y' = - 2 + 2x + 4y, y(1) = 2. Y1 = Y2 = 93 = Y4 = I| || || ||

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

### Using Euler's Method to Approximate Solutions to Differential Equations

Euler's method is a simple numerical approach to approximating solutions to ordinary differential equations (ODEs). Given the step size $ h $ and an initial value problem, the method estimates the values of the unknown function at discrete points.

#### Problem Statement
Use Euler's method with step size $ 0.5 $ to compute the approximate y-values $ y_1 $, $ y_2 $, $ y_3 $, and $ y_4 $ for the solution of the given initial-value problem:

\[ y' = -2 + 2x + 4y, \quad y(1) = 2. \]

#### Procedure

1. **Initial Value**:
- $ x_0 = 1 $
- $ y_0 = 2 $

2. **Step Size**:
- $ h = 0.5 $

3. **Equations**:
$ y' = f(x, y) = -2 + 2x + 4y $

4. **Iteration Formula**:
\[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \]

Use the iteration formula to compute $ y_1, y_2, y_3, $ and $ y_4 $.

#### Solution

1. **First Iteration**:
- $ x_0 = 1 $, $ y_0 = 2 $
- Compute $ f(x_0, y_0) $:
\[ f(1, 2) = -2 + 2(1) + 4(2) = -2 + 2 + 8 = 8 \]
- Calculate $ y_1 $:
\[ y_1 = y_0 + h \cdot f(x_0, y_0) = 2 + 0.5 \cdot 8 = 2 + 4 = 6 \]

2. **Second Iteration**:
- $ x_1 = 1.5 $, $ y_1 = 6 $
- Compute $ f(x_1, y_1) $:
\[ f(1.5, 6) = -2 + 2(1.5) +

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