dy = x – y°, y(1)= 2. dx Consider the IVP %3| Use Euler's Method to approximate y(2). Use step size h = 0.5.
dy = x – y°, y(1)= 2. dx Consider the IVP %3| Use Euler's Method to approximate y(2). Use step size h = 0.5.
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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![### Euler's Method for Approximating Solutions to Initial Value Problems
**Consider the Initial Value Problem (IVP):**
\[ \frac{dy}{dx} = x - y^2, \quad y(1) = 2. \]
**Objective:**
Use Euler’s Method to approximate \( y(2) \). Use a step size \( h = 0.5 \). (So, just two steps needed.)
In other words, find the approximate value of the solution function \( y(x) \) when \( x = 2 \).
### Procedure
You can fill in the table below:
\[
\begin{array}{|c|c|c|}
\hline
x_n & y_n & f(x_n, y_n) \\
\hline
x_0 = 1 & y_0 = 2 & \\
\hline
x_1 = 1.5 & y_1 = & \\
\hline
x_2 = 2.0 & y_2 = & \\
\hline
\end{array}
\]
### Calculation Steps
1. **Initial Point:**
- \( x_0 = 1 \)
- \( y_0 = 2 \)
- Function Value: \( f(x_0, y_0) = x_0 - y_0^2 = 1 - 2^2 = 1 - 4 = -3 \)
First Euler step:
\[ y_1 = y_0 + h \cdot f(x_0, y_0) = 2 + 0.5 \cdot (-3) = 2 - 1.5 = 0.5 \]
2. **Second Point:**
- \( x_1 = 1.5 \)
- \( y_1 = 0.5 \)
- Function Value: \( f(x_1, y_1) = x_1 - y_1^2 = 1.5 - 0.5^2 = 1.5 - 0.25 = 1.25 \)
Second Euler step:
\[ y_2 = y_1 + h \cdot f(x_1, y_1) = 0.5 + 0.5 \cdot 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F68f46f96-45b6-462c-8dc8-b421f4d1e7ff%2F1390a186-5804-460b-9151-a57ae323bcb0%2Fxsh2wazb.png&w=3840&q=75)
Transcribed Image Text:### Euler's Method for Approximating Solutions to Initial Value Problems
**Consider the Initial Value Problem (IVP):**
\[ \frac{dy}{dx} = x - y^2, \quad y(1) = 2. \]
**Objective:**
Use Euler’s Method to approximate \( y(2) \). Use a step size \( h = 0.5 \). (So, just two steps needed.)
In other words, find the approximate value of the solution function \( y(x) \) when \( x = 2 \).
### Procedure
You can fill in the table below:
\[
\begin{array}{|c|c|c|}
\hline
x_n & y_n & f(x_n, y_n) \\
\hline
x_0 = 1 & y_0 = 2 & \\
\hline
x_1 = 1.5 & y_1 = & \\
\hline
x_2 = 2.0 & y_2 = & \\
\hline
\end{array}
\]
### Calculation Steps
1. **Initial Point:**
- \( x_0 = 1 \)
- \( y_0 = 2 \)
- Function Value: \( f(x_0, y_0) = x_0 - y_0^2 = 1 - 2^2 = 1 - 4 = -3 \)
First Euler step:
\[ y_1 = y_0 + h \cdot f(x_0, y_0) = 2 + 0.5 \cdot (-3) = 2 - 1.5 = 0.5 \]
2. **Second Point:**
- \( x_1 = 1.5 \)
- \( y_1 = 0.5 \)
- Function Value: \( f(x_1, y_1) = x_1 - y_1^2 = 1.5 - 0.5^2 = 1.5 - 0.25 = 1.25 \)
Second Euler step:
\[ y_2 = y_1 + h \cdot f(x_1, y_1) = 0.5 + 0.5 \cdot 1
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