Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Euler's Method for Differential Equations**
Consider the following differential equation and initial value:
\[ y' = 2x - 3y + 1, \quad y(1) = 8; \quad y(1.2) \]
Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of equation (3) in Section 2.6:
\[ y_{n+1} = y_n + h f(x_n, y_n) \]
### First, use increment \( h = 0.1 \).
| | \( y(1) \) | **8** |
|------|--------------|-------------|
| | \( y(\_\_\_\_) \) | \[ \_\_\_\_ \] |
| | \( y(1.2) \) | **4.45** ✓ |
### Then, use increment \( h = 0.05 \). (Round your answers to four decimal places.)
| | \( y(1) \) | **8** |
|------|--------------|--------------|
| | \( y(\_\_\_\_) \) | \[ \_\_\_\_ \] |
| | \( y(1.1) \) | \[ \_\_\_\_ \] |
| | \( y(\_\_\_\_) \) | \[ \_\_\_\_ \] |
| | \( y(1.2) \) | **4.6811** ✓ |
In this exercise, we applied Euler’s method with two different increments to approximate the value of the solution at \( y(1.2) \). The results show the effectiveness of using smaller step sizes to achieve more accurate approximations.](https://content.bartleby.com/qna-images/question/25befd20-0d58-44de-b948-d9335e17825d/827395d2-86da-4534-8ea3-411cc034872a/w2zf9oa_processed.png)
Transcribed Image Text:**Euler's Method for Differential Equations**
Consider the following differential equation and initial value:
\[ y' = 2x - 3y + 1, \quad y(1) = 8; \quad y(1.2) \]
Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of equation (3) in Section 2.6:
\[ y_{n+1} = y_n + h f(x_n, y_n) \]
### First, use increment \( h = 0.1 \).
| | \( y(1) \) | **8** |
|------|--------------|-------------|
| | \( y(\_\_\_\_) \) | \[ \_\_\_\_ \] |
| | \( y(1.2) \) | **4.45** ✓ |
### Then, use increment \( h = 0.05 \). (Round your answers to four decimal places.)
| | \( y(1) \) | **8** |
|------|--------------|--------------|
| | \( y(\_\_\_\_) \) | \[ \_\_\_\_ \] |
| | \( y(1.1) \) | \[ \_\_\_\_ \] |
| | \( y(\_\_\_\_) \) | \[ \_\_\_\_ \] |
| | \( y(1.2) \) | **4.6811** ✓ |
In this exercise, we applied Euler’s method with two different increments to approximate the value of the solution at \( y(1.2) \). The results show the effectiveness of using smaller step sizes to achieve more accurate approximations.
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