**Using Euler’s Method for Estimating Values of Differential Equations**### Problem StatementUse Euler’s method with step size 0.4 to estimate $ y(0.8) $, where $ y(x) $ is the solution of the initial-value problem\[ y' = 3x + y^2, \quad y(0) = -1. \]### SolutionTo solve this problem using Euler’s method, we follow these steps:1. **Initial Setup:** - Given differential equation: $ y' = 3x + y^2 $ - Initial value: $ y(0) = -1 $ - Step size: $ h = 0.4 $ 2. **Table of Computations:**| Step | $ x $ | $ y $ | $ y' = 3x + y^2 $ | $ y_{\text{new}} = y + h \cdot y' $ ||------|--------|-------------------------|---------------------|-------------------------------------|| 0 | 0.0 | -1.000 | 3(0.0) + (-1.000)^2 = 1.000 | -1.000 + 0.4 × 1.000 = -0.600 || 1 | 0.4 | -0.600 | 3(0.4) + (-0.600)^2 = 1.800 + 0.360 = 2.160 | -0.600 + 0.4 × 2.160 = 0.264 || 2 | 0.8 | Calculated at Step 1 | - | - |### Estimate Calculation:- **Step 0:** \[ y(0) = -1 \] \[ y'(0) = 3(0) + (-1)^2 = 1 \] \[ y(0.4) \approx y(0) + h \cdot y'(0) \] \[ y(0.4) \approx -1 + 0.4 \cdot 1 = -0.6 \]- **Step 1:** \[ y'(0.4) = 3(0

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Answered: Use Euler's method with step size 0.4…

Use Euler's method with step size 0.4 to estimate y(0.8), where y(x) is the solution of the initial-value problem y' = 3x + y°, y(0) = – 1. y(0.8) =

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 for Estimating Values of Differential Equations**

### Problem Statement

Use Euler’s method with step size 0.4 to estimate $ y(0.8) $, where $ y(x) $ is the solution of the initial-value problem

\[ y' = 3x + y^2, \quad y(0) = -1. \]

### Solution

To solve this problem using Euler’s method, we follow these steps:

1. **Initial Setup:**
- Given differential equation: $ y' = 3x + y^2 $
- Initial value: $ y(0) = -1 $
- Step size: $ h = 0.4 $

2. **Table of Computations:**

| Step | $ x $ | $ y $ | $ y' = 3x + y^2 $ | $ y_{\text{new}} = y + h \cdot y' $ |
|------|--------|-------------------------|---------------------|-------------------------------------|
| 0 | 0.0 | -1.000 | 3(0.0) + (-1.000)^2 = 1.000 | -1.000 + 0.4 × 1.000 = -0.600 |
| 1 | 0.4 | -0.600 | 3(0.4) + (-0.600)^2 = 1.800 + 0.360 = 2.160 | -0.600 + 0.4 × 2.160 = 0.264 |
| 2 | 0.8 | Calculated at Step 1 | - | - |

### Estimate Calculation:

- **Step 0:**
\[ y(0) = -1 \]
\[ y'(0) = 3(0) + (-1)^2 = 1 \]
\[ y(0.4) \approx y(0) + h \cdot y'(0) \]
\[ y(0.4) \approx -1 + 0.4 \cdot 1 = -0.6 \]

- **Step 1:**
\[ y'(0.4) = 3(0

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