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Problem 6 1. Use Euler's method to find y(0.3) for the differential equation = 2 +t-y with initial value y(0) = 1 and At = 0.1. dt 2. Solve the differential equation analytically and compare the two solutions.

Problem 6 1. Use Euler's method to find y(0.3) for the differential equation = 2 +t-y with initial value y(0) = 1 and At = 0.1. dt 2. Solve the differential equation analytically and compare the two solutions.

Advanced Engineering Mathematics
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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### Problem 6 1. Use Euler’s method to find y(0.3) for the differential equation \( \frac{dy}{dt} = 2 + t - y \) with initial value \( y(0) = 1 \) and \( \Delta t = 0.1 \). 2. Solve the differential equation analytically and compare the two solutions. ### Solution: #### Table for Euler’s Method: | \( t \) | \( y \) | \( \Delta y \approx (2 + t - y) \Delta t \) | |---------|-----|--------------------------------------------| | 0 | 1 | | | 0.1 | | | | 0.2 | | | | 0.3 | | | Euler's method involves iterating the differential equation over small steps to approximate the value of the function at specific points. Given \( \Delta t = 0.1 \): 1. **Initial state \( t = 0 \) and \( y(0) = 1 \):** - \( \Delta y = (2 + 0 - 1) \cdot 0.1 = 1 \cdot 0.1 = 0.1 \) - \( y(0.1) = y(0) + \Delta y = 1 + 0.1 = 1.1 \) 2. **Next step \( t = 0.1 \) and \( y(0.1) = 1.1 \):** - \( \Delta y = (2 + 0.1 - 1.1) \cdot 0.1 = 1 \cdot 0.1 = 0.1 \) - \( y(0.2) = y(0.1) + \Delta y = 1.1 + 0.1 = 1.2 \) 3. **Next step \( t = 0.2 \) and \( y(0.2) = 1.2 \):** - \( \Delta y = (2 + 0.2 - 1.2) \cdot 0.1 = 1 \cdot 0.1 = 0.1 \) - \( y(0.3) = y(
Transcribed Image Text:### Problem 6 1. Use Euler’s method to find y(0.3) for the differential equation \( \frac{dy}{dt} = 2 + t - y \) with initial value \( y(0) = 1 \) and \( \Delta t = 0.1 \). 2. Solve the differential equation analytically and compare the two solutions. ### Solution: #### Table for Euler’s Method: | \( t \) | \( y \) | \( \Delta y \approx (2 + t - y) \Delta t \) | |---------|-----|--------------------------------------------| | 0 | 1 | | | 0.1 | | | | 0.2 | | | | 0.3 | | | Euler's method involves iterating the differential equation over small steps to approximate the value of the function at specific points. Given \( \Delta t = 0.1 \): 1. **Initial state \( t = 0 \) and \( y(0) = 1 \):** - \( \Delta y = (2 + 0 - 1) \cdot 0.1 = 1 \cdot 0.1 = 0.1 \) - \( y(0.1) = y(0) + \Delta y = 1 + 0.1 = 1.1 \) 2. **Next step \( t = 0.1 \) and \( y(0.1) = 1.1 \):** - \( \Delta y = (2 + 0.1 - 1.1) \cdot 0.1 = 1 \cdot 0.1 = 0.1 \) - \( y(0.2) = y(0.1) + \Delta y = 1.1 + 0.1 = 1.2 \) 3. **Next step \( t = 0.2 \) and \( y(0.2) = 1.2 \):** - \( \Delta y = (2 + 0.2 - 1.2) \cdot 0.1 = 1 \cdot 0.1 = 0.1 \) - \( y(0.3) = y(
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