Let y (x) be the solution of the initial value dy problem dx method with step size 0.5 to find y (2). 01 01 — O 2 B 4 = x − y, y (1) = 3. Use Euler's
Let y (x) be the solution of the initial value dy problem dx method with step size 0.5 to find y (2). 01 01 — O 2 B 4 = x − y, y (1) = 3. Use Euler's
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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![Certainly! Here is the transcription suitable for an educational website:
---
**Problem Statement:**
Let \( y(x) \) be the solution of the initial value problem
\[
\frac{dy}{dx} = x - y, \quad y(1) = 3.
\]
Use Euler's method with a step size of 0.5 to find \( y(2) \).
**Multiple Choice Options:**
- \( \frac{7}{4} \)
- 1
- \( -\frac{5}{4} \)
- 2
---
**Explanation of Euler's Method:**
Euler's method is a numerical technique used to approximate solutions to first-order ordinary differential equations (ODEs) with given initial values. It involves using an iterative process where we calculate successive points by moving a small step size along the solution's curve.
For this problem, start at the initial condition \( y(1) = 3 \) and compute successive values of \( y \) using the formula:
\[
y_{n+1} = y_n + h \cdot f(x_n, y_n)
\]
where \( h \) is the step size, and \( f(x, y) = x - y \).
**Steps to Solve:**
1. Calculate \( y(1.5) \)
2. Calculate \( y(2.0) \)
This iterative procedure will provide the approximate value of \( y(2) \) based on the initial value problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e71e459-267b-4c65-984a-67009d5ae993%2F88e8e105-d273-4401-972b-d090617361c6%2Fx00jcor_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcription suitable for an educational website:
---
**Problem Statement:**
Let \( y(x) \) be the solution of the initial value problem
\[
\frac{dy}{dx} = x - y, \quad y(1) = 3.
\]
Use Euler's method with a step size of 0.5 to find \( y(2) \).
**Multiple Choice Options:**
- \( \frac{7}{4} \)
- 1
- \( -\frac{5}{4} \)
- 2
---
**Explanation of Euler's Method:**
Euler's method is a numerical technique used to approximate solutions to first-order ordinary differential equations (ODEs) with given initial values. It involves using an iterative process where we calculate successive points by moving a small step size along the solution's curve.
For this problem, start at the initial condition \( y(1) = 3 \) and compute successive values of \( y \) using the formula:
\[
y_{n+1} = y_n + h \cdot f(x_n, y_n)
\]
where \( h \) is the step size, and \( f(x, y) = x - y \).
**Steps to Solve:**
1. Calculate \( y(1.5) \)
2. Calculate \( y(2.0) \)
This iterative procedure will provide the approximate value of \( y(2) \) based on the initial value problem.
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