Use Euler's method with step size 0.5 to compute the approximate y-values y1, Y2, Y3, and y4 of the solution of the initial-value problem y' = - 2 + 2x + 4y, y(1) = 2. Y1 = Y2 = 93 = Y4 = I| || || ||
Use Euler's method with step size 0.5 to compute the approximate y-values y1, Y2, Y3, and y4 of the solution of the initial-value problem y' = - 2 + 2x + 4y, y(1) = 2. Y1 = Y2 = 93 = Y4 = I| || || ||
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 to Approximate Solutions to Differential Equations
Euler's method is a simple numerical approach to approximating solutions to ordinary differential equations (ODEs). Given the step size \( h \) and an initial value problem, the method estimates the values of the unknown function at discrete points.
#### Problem Statement
Use Euler's method with step size \( 0.5 \) to compute the approximate y-values \( y_1 \), \( y_2 \), \( y_3 \), and \( y_4 \) for the solution of the given initial-value problem:
\[ y' = -2 + 2x + 4y, \quad y(1) = 2. \]
#### Procedure
1. **Initial Value**:
- \( x_0 = 1 \)
- \( y_0 = 2 \)
2. **Step Size**:
- \( h = 0.5 \)
3. **Equations**:
\( y' = f(x, y) = -2 + 2x + 4y \)
4. **Iteration Formula**:
\[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \]
Use the iteration formula to compute \( y_1, y_2, y_3, \) and \( y_4 \).
#### Solution
1. **First Iteration**:
- \( x_0 = 1 \), \( y_0 = 2 \)
- Compute \( f(x_0, y_0) \):
\[ f(1, 2) = -2 + 2(1) + 4(2) = -2 + 2 + 8 = 8 \]
- Calculate \( y_1 \):
\[ y_1 = y_0 + h \cdot f(x_0, y_0) = 2 + 0.5 \cdot 8 = 2 + 4 = 6 \]
2. **Second Iteration**:
- \( x_1 = 1.5 \), \( y_1 = 6 \)
- Compute \( f(x_1, y_1) \):
\[ f(1.5, 6) = -2 + 2(1.5) +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b092f3-63db-4d93-982f-67d0473d8e68%2F07f544e2-c3bc-4c03-aeec-a8f2539b0ad0%2Furjq76_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Using Euler's Method to Approximate Solutions to Differential Equations
Euler's method is a simple numerical approach to approximating solutions to ordinary differential equations (ODEs). Given the step size \( h \) and an initial value problem, the method estimates the values of the unknown function at discrete points.
#### Problem Statement
Use Euler's method with step size \( 0.5 \) to compute the approximate y-values \( y_1 \), \( y_2 \), \( y_3 \), and \( y_4 \) for the solution of the given initial-value problem:
\[ y' = -2 + 2x + 4y, \quad y(1) = 2. \]
#### Procedure
1. **Initial Value**:
- \( x_0 = 1 \)
- \( y_0 = 2 \)
2. **Step Size**:
- \( h = 0.5 \)
3. **Equations**:
\( y' = f(x, y) = -2 + 2x + 4y \)
4. **Iteration Formula**:
\[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \]
Use the iteration formula to compute \( y_1, y_2, y_3, \) and \( y_4 \).
#### Solution
1. **First Iteration**:
- \( x_0 = 1 \), \( y_0 = 2 \)
- Compute \( f(x_0, y_0) \):
\[ f(1, 2) = -2 + 2(1) + 4(2) = -2 + 2 + 8 = 8 \]
- Calculate \( y_1 \):
\[ y_1 = y_0 + h \cdot f(x_0, y_0) = 2 + 0.5 \cdot 8 = 2 + 4 = 6 \]
2. **Second Iteration**:
- \( x_1 = 1.5 \), \( y_1 = 6 \)
- Compute \( f(x_1, y_1) \):
\[ f(1.5, 6) = -2 + 2(1.5) +
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