Use the Euler's Method to approximate the solution of the equation. Complete the table. dy dx x+2y², y(0) = 2, Ax = 0.05, n = = 5 (number of points)
Use the Euler's Method to approximate the solution of the equation. Complete the table. dy dx x+2y², y(0) = 2, Ax = 0.05, n = = 5 (number of points)
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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![Sure! Below is a transcription suitable for an educational website:
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**Using Euler's Method to Approximate Solutions of a Differential Equation**
We are tasked with approximating the solution of the following differential equation using Euler's Method:
\[ \frac{dy}{dx} = x + 2y^2 \]
With the initial condition given as:
\[ y(0) = 2 \]
And the parameters specified as:
- \( \Delta x = 0.05 \)
- \( n = 5 \) (number of points)
**Euler's Method Formula:**
\[ y_n = y_{n-1} + \Delta x \cdot f(x_{n-1}, y_{n-1}) \]
**Table for Euler's Method Approximation:**
| \( n \) | \( x_n \) | \( y_n \) |
|---------|-----------|-----------|
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
**Instructions:**
1. Use the initial condition to fill in the values at \( n = 0 \).
2. Apply Euler's Method iteratively to compute \( y_n \) for each subsequent \( n \).
3. Complete the table by calculating the approximate values of \( y_n \) at discrete points \( x_n \).
This process aids in understanding how numerical methods can be used to solve differential equations by approximating solutions at discrete points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d410cd6-8fc6-4d13-8ff0-7a43ba3056bc%2F117d3891-3be8-4362-9323-376e94a29e59%2Ff9ghqqs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Sure! Below is a transcription suitable for an educational website:
---
**Using Euler's Method to Approximate Solutions of a Differential Equation**
We are tasked with approximating the solution of the following differential equation using Euler's Method:
\[ \frac{dy}{dx} = x + 2y^2 \]
With the initial condition given as:
\[ y(0) = 2 \]
And the parameters specified as:
- \( \Delta x = 0.05 \)
- \( n = 5 \) (number of points)
**Euler's Method Formula:**
\[ y_n = y_{n-1} + \Delta x \cdot f(x_{n-1}, y_{n-1}) \]
**Table for Euler's Method Approximation:**
| \( n \) | \( x_n \) | \( y_n \) |
|---------|-----------|-----------|
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
**Instructions:**
1. Use the initial condition to fill in the values at \( n = 0 \).
2. Apply Euler's Method iteratively to compute \( y_n \) for each subsequent \( n \).
3. Complete the table by calculating the approximate values of \( y_n \) at discrete points \( x_n \).
This process aids in understanding how numerical methods can be used to solve differential equations by approximating solutions at discrete points.
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Step 1: Explaining Given question
Given y′=x+2y2, y(0)=2,
Euler method
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