Use Euler's method to approximate y(1.9). Start with step size h=0.1, and then use successively smaller step sizes (h = 0.01, 0.001, 0.0001, etc.) until successive approximate solution values at x= 1.9 agree rounded off to two decimal places. y' =x? +y? - 2, y(0)= 0

Use Euler's method to approximate y(1.9). Start with step size h=0.1, and then use successively smaller step sizes (h = 0.01, 0.001, 0.0001, etc.) until successive approximate solution values at x= 1.9 agree rounded off to two decimal places. y' =x? +y? - 2, y(0)= 0

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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Question

Use Euler's method to approximate y(1.9). Start with step size h = 0.1, and then use successively smaller step sizes (h = 0.01, 0.001, 0.0001, etc.)
until successive approximate solution values at x = 1.9 agree rounded off to two decimal places.
y' =x? +y? - 2, y(0) = 0
The approximate solution values at x = 1.9 begin to agree rounded off to two decimal places between h= 0.001 andh= 0.0001. So, a good
approximation of y(1.9) is 0.11.

Expert Solution

Step 1

Euler's Method:

Let the initial value problem is given as

$y' = f (x, y), y (x_{0}) = y_{0}$ .

Then the iteration formula is given by

$y_{n + 1} = y_{n} + h f (x_{n}, y_{n})$

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