Consider the initial-value problem y'= y, y(0) - 7. (a) Use Euler's method with each of the following step sizes to estimate the value of y(0.4). (1) h=0.4 y(0.4) - (i) = 0.2 y(0.4) = (iii) = 0.1 y(0.4) =
Consider the initial-value problem y'= y, y(0) - 7. (a) Use Euler's method with each of the following step sizes to estimate the value of y(0.4). (1) h=0.4 y(0.4) - (i) = 0.2 y(0.4) = (iii) = 0.1 y(0.4) =
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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![### Example Euler's Method for Estimating y(0.4)
Consider the initial-value problem \( y' = y \), \( y(0) = 7 \).
#### (a) Use Euler’s method with each of the following step sizes to estimate the value of \( y(0.4) \).
1. **\( h = 0.4 \)**\
\( y(0.4) = \) [Input Box]
2. **\( h = 0.2 \)**\
\( y(0.4) = \) [Input Box]
3. **\( h = 0.1 \)**\
\( y(0.4) = \) [Input Box]
#### (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 \( y = 7e^x \), \( 0 \le x \le 0.4 \), 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.
The estimates are [Drop-down Menu: underestimates/overestimates].
#### (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).
- **\( h = 0.4 \) error**: \[ \( \left| \text{exact value} - \text{approximate value} \right| = \) [Input Box] \]
- **\( h = 0.2 \) error**: \[ \( \left| \text{exact value} - \text{approximate value} \right| = \) [Input Box] \]
- **\( h = 0.1 \) error**: \[ \( \left| \text{exact value} - \text{approximate value} \right| = \) [Input Box] \]
#### What happens to the error each time the step size is halved?
Each time the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad29c243-17bc-495f-9a75-f51d725c2e92%2Ff64d10ea-49e5-4608-9aa3-11378143c028%2Fpufob7p_processed.png&w=3840&q=75)
Transcribed Image Text:### Example Euler's Method for Estimating y(0.4)
Consider the initial-value problem \( y' = y \), \( y(0) = 7 \).
#### (a) Use Euler’s method with each of the following step sizes to estimate the value of \( y(0.4) \).
1. **\( h = 0.4 \)**\
\( y(0.4) = \) [Input Box]
2. **\( h = 0.2 \)**\
\( y(0.4) = \) [Input Box]
3. **\( h = 0.1 \)**\
\( y(0.4) = \) [Input Box]
#### (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 \( y = 7e^x \), \( 0 \le x \le 0.4 \), 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.
The estimates are [Drop-down Menu: underestimates/overestimates].
#### (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).
- **\( h = 0.4 \) error**: \[ \( \left| \text{exact value} - \text{approximate value} \right| = \) [Input Box] \]
- **\( h = 0.2 \) error**: \[ \( \left| \text{exact value} - \text{approximate value} \right| = \) [Input Box] \]
- **\( h = 0.1 \) error**: \[ \( \left| \text{exact value} - \text{approximate value} \right| = \) [Input Box] \]
#### What happens to the error each time the step size is halved?
Each time the
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