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) =
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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![**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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc700bc18-61e4-4e04-a31d-52f10b21c2d0%2F5831f478-cc66-4c1d-89e4-eb24dcb732dd%2Fjp0dyw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**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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