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 Euler’s method with step size h = 0.1 to approximate the solution to the initial value problem
$y' = x - y^{2}$ , y (1) = 0 at the points $x = 1.1, 1.2, 1.3, 1.4 and 1.5$ .

$x_{n}$	1.1	1.2	1.3	1.4	1.5
$y_{n}$	0.1	0.209	0.325	0.444	0.564

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Write the recursive formula

For the solution use the Euler’s formula $y_{n + 1} = y_{n} + h . f (x_{n}, y_{n})$

Apply recursive formula

We have, $f (x, y) = x - y^{2}, x_{0} = 1, y_{0} = 0, h = 0.1$

Then, $y_{n + 1} = y_{n} + h . f (x_{n}, y_{n}) = y_{n} + (0.1) (x - y^{2})$

Put n=0 to find y1

Now, find the value of $y_{1}$

$y_{1} = y_{0} + (0.1) (x_{0} - y_{0}^{2}) = 0 + (0.1) (1) = 0.1$

Hence, the value of $y_{1} = 0.1$ when $x_{1} = 1.1$

Put n=1 to find y2

The value of $y_{2}$ is

$y_{2} = y_{1} + (0.1) (x_{1} - y_{1}^{2}) = 0.1 + (0.1) (1.1 - 0.01) = 0.1 + 0.109 = 0.209$

Consequently, the value is $y_{2} = 0.209$ when $x_{2} = 1.2$

Put n=2 to find y3

Now the value of $y_{3}$

$y_{3} = y_{2} + (0.1) (x_{2} - y_{2}^{2}) = 0.209 + (0.1) (1.2 - 0.043) = 0.209 + 0.116 = 0.325$

So, the value of $y_{3} = 0.325$ when $x_{3} = 1.3$

Put n=3 to find y4

The value of $y_{4}$ is

$y_{4} = y_{3} + (0.1) (x_{3} - y_{3}^{2}) = 0.325 + (0.1) (1.3 - 0.106) = 0.325 + 0.119 = 0.444$

Thus, the value is $y_{4} = 0.444$ when $x_{4} = 1.4$

Put n=4 to find y5

The value of $y_{5}$ is

$y_{5} = y_{4} + (0.1) (x_{4} - y_{4}^{2}) = 0.444 + (0.1) (1.4 - 0.197) = 0.444 + 0.12 = 0.564$

Therefore, the value is $y_{5} = 0.564$ when $x_{5} = 1.5$

Therefore the solution is

role="math" localid="1663939345323" $x_{n}$	1.1	1.2	1.3	1.4	1.5
$y_{n}$	0.1	0.209	0.325	0.444	0.564

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Use Euler’s method with step size h = 0.1 to approximate the solution to the initial value problem
$y' = x - y^{2}$ , y (1) = 0 at the points $x = 1.1, 1.2, 1.3, 1.4 and 1.5$ .

Step by Step Solution

Write the recursive formula

Apply recursive formula

Put n=0 to find y1

Put n=1 to find y2

Put n=2 to find y3

Put n=3 to find y4

Put n=4 to find y5

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Use Euler’s method with step size h = 0.1 to approximate the solution to the initial value problem y'=x-y2, y (1) = 0 at the points x=1.1, 1.2, 1.3, 1.4 and 1.5 .

Step by Step Solution

Write the recursive formula

Apply recursive formula

Put n=0 to find y1

Put n=1 to find y2

Put n=2 to find y3

Put n=3 to find y4

Put n=4 to find y5

Most popular questions for Math Textbooks

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Pure Maths

Use Euler’s method with step size h = 0.1 to approximate the solution to the initial value problem
$y' = x - y^{2}$ , y (1) = 0 at the points $x = 1.1, 1.2, 1.3, 1.4 and 1.5$ .