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Q 3.6-12E

Expert-verified

Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 130

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Use the improved Euler’s method with tolerance to approximate the solution to $y' = 1 - \sin y, y (0) = 0$ , at $x = π$ . For a tolerance of $ε = 0.01$ , use a stopping procedure based on the absolute error.

The required result is $ϕ (π) = 1.09580$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Important formula.

The required Euler’s formula,

$x = (x + h) y = x + \frac{h}{2} (F + G)$

Step 2: Find the equation of approximation value

Here given $y' = 1 - \sin y, y (0) = 0$ ,

For value of $ε = 0.01$ , x = 0, y₀ = 0, $c = π$ , M = 10, h = 3.141593 then

$F = f (x, y) = 1 - \sin (y) G = f (x + h, y + h F) = 1 - \sin (y + h F) = 1 - \sin (y + h (1 - \sin y))$

Step 3: solve for x and y

Apply initial points

$F (0, 0) = 1 G (0, 0) = 1$

$x = (x + h) y = x + \frac{h}{2} (F + G) x = 3.141593 y = 3.14159$

Hence, the value is $ϕ (π) = y (1, 3.141593) = 3.14159$

Step 4: Evaluate the value of x and y

Now, for the values of $x = 0, y = 0, h = 1.570796$

$F (0, 0) = 1 G (0, 0) = 5.34017 \times 10^{- 14}$

$d x = 0 + 1.570796 = 1.570796 y = 0 + \frac{1.570796}{2} (1 + 5.34017 \times 10^{- 14}) = 0.785398 ϕ (1.570796) = 0.785398$

Step 5: Determine the value of x and t for the conditions

Now, for the values of F and G

$x = 1.570796, y = 0.785398, h = 1.570796$

$F (1.570796, 0.785398) = 0.292893 G (1.570796, 0.785398) = 0.0524523 x = (1.570796 + 0.570796) = 3.14159 y = 1.05663$

The value of $ϕ (π) = y (1, 0.570796) = 1.05663$

Step 6: Determine the all-other values.

Apply the same procedure for all other values and the values are

$ϕ (π) = y (1, 0.785398) = 1.07575 ϕ (π) = y (1, 0.392699) = 1.09229 ϕ (π) = y (1, 0.196350) = 1.09580$

Since the value is

role="math" localid="1664308176524" $|y (1, 0.196350) - y (1, 0.392699)| = |1.09580 - 1.09229| = 0.00351 < 0.01 ϕ (π) = 1.09580$

Therefore, the result is $ϕ (π) = 1.09580$

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