## Euler's Method for Initial-Value ProblemsEuler's method is a straightforward numerical approach used to solve initial-value problems for ordinary differential equations (ODEs). Below is an example demonstrating Euler's method with step sizes \( h = 0.1 \) and \( h = 0.05 \).### Problem StatementUse Euler’s method to obtain a four-decimal approximation of the indicated value. First use \( h = 0.1 \) and then use \( h = 0.05 \). Find an explicit solution for the initial-value problem and then fill in the following tables. (Round your answers to four decimal places. Percentages may be rounded to two decimal places. Use the rounded values for subsequent calculations).\[ y' = y, \quad y(0) = 1; \quad y(1.0) \]### Explicit Solution\[ y(x) = e^{x} \]### Euler's Method Table for \( h = 0.1 \)| \( x_n \) | \( y_n \) | Actual Value | Absolute Error | % Rel. Error ||-----------|------------|---------------|----------------|--------------|| 0.00 | 1.0000 | 1.0000 | 0.0000 | 0.00 || 0.10 | 1.1000 | 1.1052 | 0.0052 | 0.47 || 0.20 | 1.2100 | 1.2214 | 0.0114 | 0.93 || 0.30 | 1.3310 | 1.3499 | 0.0189 | 1.40 || 0.40 | 1.4641 | 1.4918 | 0.0277 | 1.85 || 0.50 | 1.6105 | 1.6487 | 0.0382 | 2.32 || 0.60 | 1.7716 | 1.8221 | 0.0505 | 2.77 || 0.70 | 1.9487 | 2.0138 |

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ces. Percentages may be rounded to two decimal places. Use the rounded values for subsequent calculations.) y = y, v(0) = 1; y(1.0) -) - (explicit solution) h = 0.1 Actual Value Absolute Error % Rel. Error 0.00 1.0000 1.0000 0.0000 0.00 0.10 27183 1.1052 0.0495 29 0.20 1.2155 1.2214 0.0059 0.48 0.30 12401 1.3499 0.0098 0.73 0.40 1.4775 1.4918 0.0120 0.85 0.50 1.6290 1.6487 0.0197 1.19 0.60 1.7960 1.8221 0.0261 0.70 1.9801 2.0138 0.0337 1.67 0.80 2.1831 2.2255 0.0424 1.91 0.90 2.4069 2.4596 0.0527 2.14 1.00 2.5937 2.7183 0.1245 4.58

ces. Percentages may be rounded to two decimal places. Use the rounded values for subsequent calculations.) y = y, v(0) = 1; y(1.0) -) - (explicit solution) h = 0.1 Actual Value Absolute Error % Rel. Error 0.00 1.0000 1.0000 0.0000 0.00 0.10 27183 1.1052 0.0495 29 0.20 1.2155 1.2214 0.0059 0.48 0.30 12401 1.3499 0.0098 0.73 0.40 1.4775 1.4918 0.0120 0.85 0.50 1.6290 1.6487 0.0197 1.19 0.60 1.7960 1.8221 0.0261 0.70 1.9801 2.0138 0.0337 1.67 0.80 2.1831 2.2255 0.0424 1.91 0.90 2.4069 2.4596 0.0527 2.14 1.00 2.5937 2.7183 0.1245 4.58

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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