(c) The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value Fy(0.4), namely, 7e0.4. (Round your answers to four decimal places.) h = 0.4 error = |(exact value) - (approximate value)| = h = 0.2 error = |(exact value) - (approximate value) | = h = 0.1 error = |(exact value) - (approximate value) | = What happens to the error each time the step size is halved? Each time the step size is halved, the error estimate appears to be --Select--- ✓(approximately).
(c) The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value Fy(0.4), namely, 7e0.4. (Round your answers to four decimal places.) h = 0.4 error = |(exact value) - (approximate value)| = h = 0.2 error = |(exact value) - (approximate value) | = h = 0.1 error = |(exact value) - (approximate value) | = What happens to the error each time the step size is halved? Each time the step size is halved, the error estimate appears to be --Select--- ✓(approximately).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
I need help with C) please
![# Euler's Method Application in Solving Initial-Value Problems
Consider the initial-value problem \( y' = y \), \( y(0) = 7 \).
## Part (a)
**Objective:** Use Euler’s method with each of the following step sizes to estimate the value of \( y(0.4) \).
1. **Step Size (\(h\)) = 0.4**
\( y(0.4) = 9.8 \) ✓
2. **Step Size (\(h\)) = 0.2**
\( y(0.4) = 10.08 \) ✓
3. **Step Size (\(h\)) = 0.1**
\( y(0.4) = 10.2487 \) ✓
## Part (b)
We know that the exact solution of the initial-value problem in part (a) is \( y = 7e^x \). Draw, as accurately as you can, the graph of \( y = 7e^x \), \( 0 \leq x \leq 0.4 \), together with the Euler approximations using the step sizes in part (a). Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates.
```plaintext
The estimates are [Select an option].
```
## Part (c)
The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value of \( y(0.4) \), namely, \( 7e^{0.4} \).
(Round your answers to four decimal places.)
1. **Step Size (\(h\)) = 0.4**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
2. **Step Size (\(h\)) = 0.2**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
3. **Step Size (\(h\)) = 0.1**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
```plaintext
What happens to the error each time the step size is halved?
Each time the step size is halved, the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad29c243-17bc-495f-9a75-f51d725c2e92%2Fb81ba7cf-48e2-496f-8b3c-70e871404b4a%2Fe50var_processed.png&w=3840&q=75)
Transcribed Image Text:# Euler's Method Application in Solving Initial-Value Problems
Consider the initial-value problem \( y' = y \), \( y(0) = 7 \).
## Part (a)
**Objective:** Use Euler’s method with each of the following step sizes to estimate the value of \( y(0.4) \).
1. **Step Size (\(h\)) = 0.4**
\( y(0.4) = 9.8 \) ✓
2. **Step Size (\(h\)) = 0.2**
\( y(0.4) = 10.08 \) ✓
3. **Step Size (\(h\)) = 0.1**
\( y(0.4) = 10.2487 \) ✓
## Part (b)
We know that the exact solution of the initial-value problem in part (a) is \( y = 7e^x \). Draw, as accurately as you can, the graph of \( y = 7e^x \), \( 0 \leq x \leq 0.4 \), together with the Euler approximations using the step sizes in part (a). Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates.
```plaintext
The estimates are [Select an option].
```
## Part (c)
The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value of \( y(0.4) \), namely, \( 7e^{0.4} \).
(Round your answers to four decimal places.)
1. **Step Size (\(h\)) = 0.4**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
2. **Step Size (\(h\)) = 0.2**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
3. **Step Size (\(h\)) = 0.1**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
```plaintext
What happens to the error each time the step size is halved?
Each time the step size is halved, the
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
